Question

The quantity, Q, of a certain product manufactured depends on the quantity of labor, L, and of capital, K, used according to the function

Q = 900L¹/²K²/³.
Labor costs $100 per unit and capital costs $200 per unit. What combination of labor and capital should be used to produce 36,000 units of the goods at minimum cost? (Round your answers to the nearest whole number.)
K = L = What is that minimum cost? (Round your answer to the nearest whole number.)

Answer 1

The best production method when labor is $100/unit and capital is $400/unit is Method 1, with a total cost of $9000. If the cost of labor increases to $200/unit, Method 1 is still the cheapest option, now at a total cost of $14000.

Step-by-step explanation:

To determine the best production method when labor costs $100/unit and capital costs $400/unit, we calculate the total cost for each method:

• Method 1: Total cost = (50 units of labor × $100/unit) + (10 units of capital × $400/unit) = $5000 + $4000 = $9000
• Method 2: Total cost = (20 units of labor × $100/unit) + (40 units of capital × $400/unit) = $2000 + $16000 = $18000
• Method 3: Total cost = (10 units of labor × $100/unit) + (70 units of capital × $400/unit) = $1000 + $28000 = $29000

Method 1 is the cheapest with a total cost of $9000.

If the cost of labor rises to $200/unit, we recalculate:

• Method 1: Total cost = (50 units of labor × $200/unit) + (10 units of capital × $400/unit) = $10000 + $4000 = $14000
• Method 2: Total cost = (20 units of labor × $200/unit) + (40 units of capital × $400/unit) = $4000 + $16000 = $20000
• Method 3: Total cost = (10 units of labor × $200/unit) + (70 units of capital × $400/unit) = $2000 + $28000 = $30000

With the increased labor cost, Method 1 remains the cheapest at $14000, so the company should still use Method 1.

Answer 2

The best production method given labor cost of $100/unit and capital cost of $400/unit is Method 1, with a cost of $9000. If the cost of labor increases to $200/unit, Method 1 remains the most cost-effective, but the total rises to $14000.

Step-by-step explanation:

To determine the best production method based on cost when the cost of labor is $100 per unit and the cost of capital is $400 per unit, we need to calculate the total cost of each method. Applying the cost values to the given methods, we find that:

Method 1: (50 units of labor × $100/unit) + (10 units of capital × $400/unit) = $5000 + $4000 = $9000

Method 2: (20 units of labor × $100/unit) + (40 units of capital × $400/unit) = $2000 + $16000 = $18000

Method 3: (10 units of labor × $100/unit) + (70 units of capital × $400/unit) = $1000 + $28000 = $29000

Therefore, the best production method is Method 1, with the lowest cost of $9000.

If the cost of labor rises to $200 per unit, we recalculate:

Method 1: (50 units of labor × $200/unit) + (10 units of capital × $400/unit) = $10000 + $4000 = $14000

Method 2: (20 units of labor × $200/unit) + (40 units of capital × $400/unit) = $4000 + $16000 = $20000

Method 3: (10 units of labor × $200/unit) + (70 units of capital × $400/unit) = $2000 + $28000 = $30000

In this scenario, the best method based on the increased labor cost remains Method 1, but now with a higher total cost of $14000.