Question

A small company that shovels sidewalks and driveways has 100 homes signed up for its services this winter. It can use various combinations of capital and labor: lots of labor with hand shovels, less labor with snow blowers, and still less labor with a pickup truck that has a snowplow on front. To summarize, the method choices are:
• Method 1: 50 units of labor, 10 units of capital
• Method 2: 20 units of labor, 40 units of capital
• Method 3: 10 units of labor, 70 units of capital

If hiring labor for the winter costs $100/unit and a unit of capital costs $400, what production method should be chosen? What method should be chosen if the cost of labor rises to $200/unit?

Answer 1

Answer:

(i) Method 1

(ii) Method 1

Explanation:

(i) Let L denote units of labor and C denote units of capital. Initially, the cost function for each method is given by the following expression:

$M_i = \ 100*L_i+\ 400*C_i$

Applying the given values for all three methods, the one with the lowest cost should be picked:

$M_1= \ 100*50+\ 400*10\\M_1 = \ 9,000\\M_2= \ 100*20+\ 400*40\\M_2 = \ 18,000\\M_3= \ 100*10+\ 400*70\\M_3 = \ 29,000$

Therefore, Method 1 should be picked.

(ii) If the cost of labor rises to $200/ unit:

$M_1= \ 200*50+\ 400*10\\M_1 = \ 14,000\\M_2= \ 200*20+\ 400*40\\M_2 = \ 20,000\\M_3= \ 200*10+\ 400*70\\M_3 = \ 30,000$

Method 1 is still the most cost attractive method.

Answer 2

Answer:

Method 1 should be chose, since it is still the cheapest if labor cost rises to $200/unit.

Explanation:

Total Cost = ( units * labor costs) + (capital cost * units of labor)

Total Cost for Method 1 : (50 * 100) + (10*400)

= $9,000

Total Cost for Method 2 : (20 * 100) + (40*400)

= $18,000

Total Cost for Method 3 : (10 * 100) + (70*400)

= $29,000

If  the price of labor rises to $200 then:

Total Cost for Method 1 : (50 * 200) + (10*400)

= $14,000

Total Cost for Method 2 : (20 * 200) + (40*400)

= $20,000

Total Cost for Method 3 : (10 * 200) + (70*400)

= $30,000