Let's analyze the problem and calculate the requested values from the provided mathematical function and data.

Problem Overview

The production function for the Build My Dream (BMD) factory is:

$y = -50 + 12l - 3(0.5)l^2 + 8k - (0.5)k^2 + l \cdot k$

Where:

• $l$: Number of working hours by the production line workers (hours/day).
• $k$: Number of production lines (units).
• $y$: Number of cars produced per day.

1.1 Calculations

Given:

• $l = 6$ hours/day.
• $k = 8$ production lines.

We will compute:

1. Output ($y$): The total number of cars produced.
2. Marginal Product of Labor (MPL): The additional output from one more unit of labor, keeping $k$ constant.
3. Average Product of Labor (APL): The output per working hour.

Step 1: Compute Output ($y$)

Substitute $l = 6$ and $k = 8$ into the function:

$y = -50 + 12(6) - 3(0.5)(6)^2 + 8(8) - (0.5)(8)^2 + (6)(8)$

Simplify step by step: $y = -50 + 72 - 3(0.5)(36) + 64 - (0.5)(64) + 48$ $y = -50 + 72 - 54 + 64 - 32 + 48$ $y = 148 \text{ cars/day}.$

Step 2: Compute MPL

MPL is the partial derivative of $y$ with respect to $l$: $\frac{\partial y}{\partial l} = 12 - 3(0.5)(2l) + k$

Substitute $l = 6$ and $k = 8$: $\frac{\partial y}{\partial l} = 12 - 3(0.5)(12) + 8$ $\frac{\partial y}{\partial l} = 12 - 18 + 8$ $\frac{\partial y}{\partial l} = 2 \text{ cars/hour}.$

Step 3: Compute APL

APL is the total output divided by the total working hours: $APL = \frac{y}{l}$

Substitute $y = 148$ and $l = 6$: $APL = \frac{148}{6}$ $APL \approx 24.67 \text{ cars/hour}.$

Summary of Results

1. Output ($y$): 148 cars/day.
2. Marginal Product of Labor (MPL): 2 cars/hour.
3. Average Product of Labor (APL): 24.67 cars/hour.

Would you like further explanations or a breakdown of the formulas?

Related Questions

1. What does the marginal product of labor (MPL) indicate in production economics?
2. How does the production function change if we increase the number of production lines ($k$)?
3. What happens to the APL when $l$ is significantly increased?
4. How can we calculate the marginal product of capital (MPK) for the same function?
5. What is the significance of diminishing marginal returns in this problem?

Tip

When working with production functions, always check for diminishing marginal returns by observing how changes in $l$ or $k$ affect the marginal product.