Final answer:
The optimal production method for the firm is Method 1, both before and after the increase in labor cost. Initially, Method 1 costs $9000 compared to the more expensive alternatives, and it remains the cheapest at $14000 even after the cost of labor increases.
Step-by-step explanation:
The question falls into the realm of Business, particularly focusing on production functions and cost optimization in economics. To determine the optimal amount of labor and capital that a firm should employ in the long run, we need to consider both the costs associated with hiring labor and buying capital, and the production function given for the firm.
In Method 1, the total cost would be (50 units of labor × $100/unit) + (10 units of capital × $400/unit) = $5000 + $4000 = $9000. Method 2 would cost (20 × $100) + (40 × $400) = $2000 + $16000 = $18000. Method 3 would cost (10 × $100) + (70 × $400) = $1000 + $28000 = $29000.
Comparing the costs, Method 1 is the most cost-effective initially. If the cost of labor rises to $200/unit, then the new costs would be: Method 1 = (50 × $200) + (10 × $400) = $10000 + $4000 = $14000. Thus, under these new conditions, Method 1 remains the most cost-effective approach for production.