Question

Consider a firm with the following production function: q = f(K, L) = K^2L. The rental rate (r) of a unit of capital is $200, and the daily wage (w) per worker hired is $50. What is the firm's optimal combination of capital and labor to minimize costs while producing a given level of output?

Answer 1

The firm's optimal combination of capital and labor to minimize costs is Method 1 when labor costs $100/unit and it remains optimal even when labor cost increases to $200/unit.

Step-by-step explanation:

To determine the firm's optimal combination of capital and labor to minimize costs while producing a given level of output, we need to calculate the total cost for each method of production provided and then compare them. Here is the calculation under the two different scenarios for labor costs:

• 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

When the cost of labor is $100/unit, Method 1 is the cheapest at $9000 total cost. If the cost of labor rises to $200/unit, the new costs will be:

• 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

With the increased labor cost, Method 1 remains the most cost-effective option.