Question

You manage a plant that mass-produces engines by teams of workers using assembly machines. The technology is summarized by the production function \\[ q=5 K L \\] where \(q\) is the number of engines per week, \(K\) is the number of assembly machines, and \(L\) is the number of labor teams. Each assembly machine rents for \(r=\$ 10,000\) per week, and each team costs \(w=\$ 5000\) per week. Engine costs are given by the cost of labor teams and machines, plus \(\$ 2000\) per engine for raw materials. Your plant has a fixed installation of 5 assembly machines as part of its design. a. What is the cost function for your plant-namely, how much would it cost to produce \(q\) engines? What are average and marginal costs for producing \(q\) engines? How do average costs vary with output? b. How many teams are required to produce 250 engines? What is the average cost per engine? c. You are asked to make recommendations for the design of a new production facility. What capital/ labor \((K / L)\) ratio should the new plant accommodate if it wants to minimize the total cost of producing at any level of output \(q ?\)

Short answer

This problem involves the calculation of a cost function, the average cost and the marginal costs based on a specific production function. It also involves determining an optimal capital-labor ratio for a new production facility and estimating the number of labor teams required to produce 250 engines, along with the average cost per engine.

Step-by-step solution

Calculate Cost Function

First, calculate the cost function, C(q), which should provide the total cost to produce \(q\) engines. We will estimate this cost function based on given machine and labor costs plus the cost of the raw materials. Since the plant has 5 machines, which costs $10,000 each per week, \(K = 5\) and \(rK = \$10,000 * 5\). Then, calculate the Labor Cost replacing \(L\) from the production function \(q=5KL\) by \(L = \dfrac{q}{5K}\) and multiplying it by the labor cost per team (w), we have: \(wL = \$5,000 * \dfrac{q}{5K}\). Finally, add the raw material cost per engine to get the cost function: \(C(q) = rK + wL + \$2,000*q\).

Calculate Average and Marginal Costs

Next, calculate the average cost (AC) by dividing the cost function by the quantity \(q\) (i.e., \(AC(q) = C(q) / q\)). Marginal cost (MC), which is the cost of producing one additional unit, is the derivative of the cost function (i.e., \(MC(q) = dC(q) / dq\)).

Determine Labor Needs for 250 Engines

Substitute \(q = 250\) into the production function to determine the number of labor teams needed to produce 250 engines. We can get an equation as \(250 = 5*K*L\). Here \(K = 5\), therefore, substitute \(K\) and solve the equation to get the value of \(L\). Calculate the average cost per engine by substituting \(q = 250\) into the average cost function.

Recommendation for New Production Facility

Here, we need to derive a general \(K/L\) ratio that would minimize the total cost for any level of output \(q\). This requires differentiating the cost function with respect to \(L\) and setting the result to zero, then solving for the \(L\). This will give the optimal capital-labor ratio \(K^*/L^*\).

Key concepts

Average Cost

The concept of average cost is essential in understanding how the total cost is distributed among the number of units produced. When we talk about average cost in the context of producing engines, this refers to how much it costs, on average, to produce one engine.

To calculate the average cost, you take the total cost function, which includes all expenses like labor, machinery, and raw materials, and divide it by the number of engines produced. In formula terms, this is expressed as:

\[ AC(q) = \frac{C(q)}{q} \]

Here, \(C(q)\) represents the total cost for producing \(q\) engines, and \(q\) is the number of engines.

• If you produce more engines, this can spread fixed costs over more units, potentially lowering the average cost.
• Understanding how the average cost changes with production levels can help in scaling production efficiently.
• It's crucial when setting pricing strategies to ensure profitability or competitive advantage in the market.

This average cost calculation gives insight into efficiency and cost management at different production levels.

Marginal Cost

Marginal cost provides insight into how costs change with an increase in production, specifically, the cost of producing one additional engine. Understanding marginal cost is useful for decision-making on whether it's beneficial to increase production.

Mathematically, marginal cost can be found using the derivative of the cost function, \(C(q)\), with respect to the quantity \(q\). This is formulated as:

\[ MC(q) = \frac{dC(q)}{dq} \]

Here are some critical points about marginal cost:

• Rising marginal costs may suggest straining resources or inefficiencies in production, compelling managers to reconsider production levels.
• Persistently high marginal costs compared to the revenue per unit may suggest it's time to rethink the pricing strategy or production process.
• Lower marginal costs can indicate that increasing production is economically advantageous, possibly leading to higher profits.

Understanding this calculation aids managers in making informed production decisions, optimizing production lines, and maintaining economic viability.

Capital-Labor Ratio

The capital-labor ratio is a critical concept when designing a new production facility. It represents the balance between the number of machines (capital) and labor teams required for efficient production.

In simpler terms, this ratio tells us how much capital is used per labor unit. From a production and cost optimization perspective, finding the right balance maximizes efficiency and minimizes costs. When formulating the ideal capital-labor ratio, consider:

• This balance can influence overall productivity and cost-effectiveness.
• A higher ratio may mean more machinery and less labor, suitable for automation-based plants.
• A lower ratio might involve more labor, ideal for processes that require human oversight or dexterity.
• The optimal ratio ensures that neither capital nor labor is underutilized, which reduces wastage and enhances productivity.

When setting up a new plant, aiming for the ideal capital-labor ratio minimizes production costs while maximizing output potential, aligning with strategic business goals.