John Mills- the propietor of Mills Engine Company, a manufacturer of mdel airplane engine-finds that it takes units of labor and units of capital to produce units of the product. If a unit of labor costs , a unit of capital costs and is bugeted for production, determine how long units should be expended on on labour and how many units should be expended on capital in order to maximize production.
A) units of labour and units of capital.
B) units of labour and units of capital.
C) units of labour and units of capital.
D) units of labour and units of capital.
E) units of labour and units of capital.

Question

John Mills- the propietor of Mills Engine Company, a manufacturer of mdel airplane engine-finds that it takes  units of labor and  units of capital to produce  units of the product. If a unit of labor costs  , a unit of capital costs  and  is bugeted for production, determine how long units should be expended on on labour and how many units should be expended on capital in order to maximize production.
A) units of labour and units of capital. 
B) units of labour and units of capital. 
C) units of labour and units of capital. 
D) units of labour and units of capital. 
E) units of labour and units of capital.

Quizplus · Accepted Answer

To maximize production, we need to find the optimal combination of labor and capital that minimizes the cost of production while using the full budget of 11eaa8b1_d99e_ec41_b1b6_e96a17fc2670_TB6026_11. We can use the formula:

Cost of production = (Cost of labor x Units of labor) + (Cost of capital x Units of capital)

Substituting the given values, we get:

Cost of production = (11eaa8b1_d99e_c52f_b1b6_796f65dca997_TB6026_11 x 11eaa8b1_d99e_770c_b1b6_efd7efc61637_TB6026_11) + (11eaa8b1_d99e_c530_b1b6_bb264eb7b96d_TB6026_11 x 11eaa8b1_d99e_9e1d_b1b6_714312f9923e_TB6026_11)

Simplifying, we get:

Cost of production = 11eaa8b1_d99f_ce8d_b1b6_aa68bdc1f906_TB6026_11

Now, we can find the partial derivative of the cost of production with respect to labor and capital:

∂Cost/∂Labor = 11eaa8b1_d99e_c52f_b1b6_796f65dca997_TB6026_11

∂Cost/∂Capital = 11eaa8b1_d99e_c530_b1b6_bb264eb7b96d_TB6026_11

Setting these partial derivatives equal to zero, we get:

11eaa8b1_d99e_c52f_b1b6_796f65dca997_TB6026_11 = λ x 11eaa8b1_d99e_c530_b1b6_bb264eb7b96d_TB6026_11

(where λ is the Lagrange multiplier)

Simplifying, we get:

λ = 11eaa8b1_d99f_1352_b1b6_1fedf7bd9684_TB6026_11

Substituting λ in either equation, we get:

11eaa8b1_d99e_c52f_b1b6_796f65dca997_TB6026_11 = 11eaa8b1_d99f_1352_b1b6_1fedf7bd9684_TB6026_11 x 11eaa8b1_d99e_c530_b1b6_bb264eb7b96d_TB6026_11

Solving for either variable, we get:

Units of labor = 11eaa8b1_d99f_af98_b1b6_f3e2b07e25b9_TB6026_11

Units of capital = 11eaa8b1_d99f_d6a9_b1b6_497fa458c45e_TB6026_11

Therefore, the optimal combination is 11eaa8b1_d99f_af98_b1b6_f3e2b07e25b9_TB6026_11 units of labor and 11eaa8b1_d99f_d6a9_b1b6_497fa458c45e_TB6026_11 units of capital, which is option D.

John Mills- the Propietor of Mills Engine Company, a Manufacturer