91Ó°ÊÓ

Amy, Bill, and Carla all mow lawns for money. Each of them operates a different lawn mower. The accompanying table shows the total cost to Amy, Bill, and Carla of mowing lawns. $$ \begin{array}{cccc} \begin{array}{c} \text { Quantity of } \\ \text { lawns mowed } \end{array} & \begin{array}{c} \text { Amy's } \\ \text { total cost } \end{array} & \begin{array}{c} \text { Bill's } \\ \text { total cost } \end{array} & \begin{array}{c} \text { Carla's } \\ \text { total cost } \end{array} \\ 0 & \$ 0 & \$ 0 & \$ 0 \\ 1 & 20 & 10 & 2 \\ 2 & 35 & 20 & 7 \\ 3 & 45 & 30 & 17 \\ 4 & 50 & 40 & 32 \\ 5 & 52 & 50 & 52 \\ 6 & 53 & 60 & 82 \end{array} $$ a. Calculate Amy's, Bill's, and Carla's marginal costs, and draw each of their marginal cost curves. b. Who has increasing marginal cost, who has decreasing marginal cost, and who has constant marginal cost?

Step by step solution

01

Calculate the marginal costs.

To find the marginal cost (MC) for each person, subtract the total cost of mowing the previous lawn (TCL-1) from the total cost of mowing the current lawn (TCL). That is, MC = TCL - TCL-1. Perform this calculation for Amy, Bill, and Carla for each quantity of lawns mowed.

02

Analyze the marginal costs.

Observe the computed marginal costs to determine who has increasing, decreasing, and constant marginal costs. An increasing MC is when the MC increases as the quantity of lawns mowed increases; a decreasing MC is when the MC decreases as the quantity of lawns mowed increases; and a constant MC is when the MC remains the same regardless of the number of lawns mowed.

03

Draw each person's marginal cost curve.

Plot the marginal cost (y-axis) against the quantity of lawns mowed (x-axis) for each person on separate graphs or on a single graph with different colors or symbols to represent each person.

04

Calculating Marginal Costs:

Refer to the table, and calculate the marginal costs for each person: - Amy: \(\operatorname{MC} = (20-0, 35-20, 45-35, 50-45, 52-50, 53-52) = (20, 15, 10, 5, 2, 1)\) - Bill: \(\operatorname{MC} = (10-0, 20-10, 30-20, 40-30, 50-40, 60-50) = (10, 10, 10, 10, 10, 10)\) - Carla: \(\operatorname{MC} = (2-0, 7-2, 17-7, 32-17, 52-32, 82-52) = (2, 5, 10, 15, 20, 30)\)

05

Analyzing Marginal Costs:

Based on the calculated marginal costs: - Amy has decreasing marginal costs, as the MC decreases as the quantity of lawns mowed increases. - Bill has constant marginal costs, as the MC remains the same regardless of the number of lawns mowed. - Carla has increasing marginal costs, as the MC increases as the quantity of lawns mowed increases.

06

Drawing Marginal Cost Curves:

To plot the marginal cost curves, set the x-axis as the number of lawns mowed and the y-axis as the marginal cost. The curve for Amy would be downward sloping, Bill's curve would be a horizontal line, and Carla's curve would be an upward-sloping line.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Decreasing Marginal Cost

Decreasing marginal cost is an interesting phenomenon and can occur when the cost to produce an additional unit, in this case mowing one more lawn, gets cheaper as you increase the number of units. For example, Amy's marginal costs demonstrate this behavior. Initially, her cost is high at $20 for the first lawn. But as she continues to mow more lawns, her marginal cost decreases. It's $15 for the second lawn, then $10, and so on.

This kind of cost pattern can happen due to various factors, such as increasing efficiency with experience or taking advantage of economies of scale.

• Efficiency gains can occur when repeated tasks become faster and easier over time.
• Economies of scale might be present when bulk buying resources or utilizing tools and resources more effectively.

Understanding this can help predict how costs might behave as production ramps up, informing decisions to enhance efficiency.

Constant Marginal Cost

Constant marginal cost is straightforward: the cost to produce one more unit remains the same, no matter how many you already produced. Bill's mowing operation reflects this perfect stability in cost. Every lawn he mows costs him consistently $10, making it a classic case of constant marginal cost.

Such a pattern often appears in scenarios where each additional unit has no associated changes in production method or resource use—

• Perhaps Bill's equipment works optimally with no wear and tear over the short run.
• It might also mean input costs—like fuel or maintenance—are consistent.

Constant marginal costs allow businesses to predict expenses accurately, making financial planning more manageable.

Increasing Marginal Cost

Increasing marginal cost happens when each additional unit becomes more expensive to produce than the last. Carla's lawn mowing costs reflect this situation. As she mows more lawns, the cost rises—from $2 for the first lawn, climbing steadily to $30 for the sixth.

This increase may arise in scenarios like:

• Resource scarcity, where more effort or material is needed for each further unit.
• Higher output might strain resources or lead to inefficiencies, like using older, less efficient equipment after the best machines are fully utilized.

Recognizing when and why these costs increase allows for strategic decisions—such as investing in better equipment to prevent cost hikes or choosing to limit production before costs outweigh benefits.