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: intensive 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 is the best production method? What method should the company use if the cost of labor rises to \(\$ 200 /\) unit?

Short answer

Initially, the most cost-effective method is Method 1 with a total cost of $9,000. When the cost of labor rises to $200 per unit, Method 1 remains the most cost-effective method with a new total cost of $14,000. However, it is crucial to recalculate costs with updated labor and/or capital costs if needed.

Step-by-step solution

Calculate the cost of each method initially.

Costs are calculated by using the formula \(cost = (labor \,units \times cost\, per\, labor\, unit) + (capital\, units \times cost\, per\, capital\, unit)\). \[ \text{Method 1 Cost} = (50 \,units \times \$100\) + (10\, units \times \$400) = \$9000 \] \[ \text{Method 2 Cost} = (20\, units \times \$100) + (40\, units \times \$400) = \$18000 \] \[ \text{Method 3 Cost} = (10\, units \times \$100) + (70\, units \times \$400) = \$29000 \] By comparison, we can see that Method 1, with a total cost of $9000, is the most cost-effective initially.

Calculate the cost of each method if the cost of labor rises to $200 per unit.

If the cost of labor rises to $200/unit, we have to recalculate the cost for each method: \[ \text{Method 1 New Cost} = (50\, units \times \$200) + (10\, units \times \$400) = \$14000 \] \[ \text{Method 2 New Cost} = (20\, units \times \$200) + (40\, units \times \$400) = \$20000 \] \[ \text{Method 3 New Cost} = (10\, units \times \$200) + (70\, units \times \$400) = \$29000 \] With the rise in labor cost, Method 1 still remains the most cost-effective, now at a cost of $14,000. However, it is important to note that if the cost of labor continues to rise, Method 1 may not remain the most cost-effective. Always recalculate costs with updated labor and/or capital costs to ensure the most cost-efficient method is chosen.

Key concepts

Cost Efficiency

Understanding cost efficiency is crucial in choosing the right production method for a company. Cost efficiency involves minimizing the expenses required to produce a given level of output, or conversely, maximizing the output for a given total cost. In our exercise, cost efficiency is calculated by comparing the total costs of different production methods that combine various amounts of labor and capital to provide services.

To demonstrate, the company can shovel sidewalks using significantly varying levels of labor and capital. With Method 1 being the least capital-intensive and Method 3 the most, the cost efficiency of each method is brought into focus when labor and capital costs are applied. The solution shows that initially, Method 1 is the most cost-efficient, indicating that for the particular set of costs given, using more labor and less capital minimizes expenses. However, changes in the costs of labor or capital would require reevaluation to ensure the selected method remains the most cost-efficient.

Labor and Capital Trade-off

The labor and capital trade-off is a key consideration for businesses deciding on the right production method. It revolves around the substitution of labor (human resources) for capital (machines or equipment) in the production process. An optimal balance between these two inputs can significantly affect overall costs and operational efficiency.

In the context of our example, the company has three methods to choose from, each with a different ratio of labor to capital. Method 1 relies heavily on labor, while Method 3 relies more on capital. The decision of which method to use often depends on the relative costs of labor and capital. As shown in the solution, when the cost of labor increases, the comparison of total costs among the methods highlights the significant impact that changes in input prices can have. Even as labor costs rise, however, Method 1 remains the most economical choice, underlining the value of reevaluating the labor-capital ratio in response to fluctuating input prices.

Production Method Selection

When selecting the most appropriate production method, businesses must weigh several factors beyond straightforward cost efficiency. The decision involves analyzing the scale of operations, the flexibility of input usage, quality considerations, and responsiveness to changing market conditions.

For our small company servicing 100 homes, Method 1 is initially the most cost-efficient, but this may change with varying circumstances such as increasing labor costs or the need for quicker snow removal after heavy snowfall. A thorough analysis would also consider the impact on service quality and customer satisfaction, as well as the agility of shifting between methods in response to sudden changes in weather patterns or market demands. In conclusion, the optimal production method is a dynamic decision where cost, quality, flexibility, and scalability need to be balanced in line with the company's strategic objectives and operational constraints.