Question

The following table shows data about quantity produced and total cost of production in the long run for a given firm: Find the long-run marginal cost and the long-run average cost faced by the firm. On a graph, plot the \(L M C\) and \(L A C\) curves. Explain why the \(L M C\) curve cuts the \(L A C\) curve from below.

Short answer

Compute LMC and LAC, plot them, LMC cuts LAC from below.

Step-by-step solution

Understanding the Problem

We are given a table that includes the quantity produced and the total cost of production for a firm. Our task is to find the long-run marginal cost (LMC) and the long-run average cost (LAC), then plot these on a graph and discuss their relationship.

Calculate the Long-Run Average Cost (LAC)

The long-run average cost for each output level is calculated by dividing the total cost by the quantity produced. If the table is not included, assume you have values to compute using: \[ \text{LAC} = \frac{\text{Total Cost}}{\text{Quantity Produced}} \] Compute LAC for each data point.

Calculate the Long-Run Marginal Cost (LMC)

The long-run marginal cost is the change in total cost divided by the change in output. Using sequential pairs of quantity and cost, calculate:\[ \text{LMC} = \frac{\Delta \text{Total Cost}}{\Delta \text{Quantity}} \]Compute LMC between each consecutive quantity level.

Plot the LMC and LAC Curves

Create a graph with the quantity produced on the horizontal axis and costs (both LMC and LAC) on the vertical axis. Plot the computed LMC and LAC values and join them to form curves.

Analyze the Curves Intersection

The intersection of LMC and LAC curves is important because it is the point where the additional cost of producing one more unit is exactly equal to the average cost of the units produced so far. LMC will cut the LAC curve from below due to the property that average costs decrease until marginal costs surpass them.

Key concepts

Long-Run Marginal Cost (LMC)

In the realm of economics, understanding the concept of Long-Run Marginal Cost (LMC) is crucial for analyzing how a firm’s production decisions evolve over time in response to changing costs. The LMC represents the additional cost incurred when one additional unit of output is produced in the long run, taking into account all inputs used by the firm are variable, enabling the firm to adjust and optimize their production processes fully.
The calculation of LMC is straightforward: it’s the change in total cost divided by the change in quantity, typically expressed as \( \text{LMC} = \frac{\Delta \text{Total Cost}}{\Delta \text{Quantity}} \). This measure helps firms determine the breakeven point where producing further units will begin to yield diminishing returns compared to their costs. Thus, it becomes a vital tool in ensuring firms operate efficiently and competitively in their respective markets.

Long-Run Average Cost (LAC)

When we talk about Long-Run Average Cost (LAC), we are looking at the average cost per unit of output when all costs are variable, reflecting an optimal level of production capacity over time. LAC is pivotal because it helps firms assess at what production level they can achieve the lowest cost per unit.
To calculate LAC, you divide the total cost of production by the quantity produced, represented as \( \text{LAC} = \frac{\text{Total Cost}}{\text{Quantity Produced}} \). This metric provides insight into how efficiently a firm utilizes its resources over an extended period. An essential aspect of LAC is its shape on a graph - usually U-shaped. This mirrors how costs decline as output increases initially (due to economies of scale) and then may rise again when diseconomies of scale set in as the firm overextends its resources.

Cost Curves Analysis

Analyzing cost curves offers a broader understanding of how a firm's costs behave with varying production levels. The relationship between Long-Run Marginal Cost (LMC) and Long-Run Average Cost (LAC) can be visualized through their respective curves on a graph. As quantities produced increase, these curves show distinct trends and intersections that hold significant implications for the firm's production strategy.

• The LMC curve often intersects the LAC curve at its lowest point. This is because marginal costs impact average costs, setting the direction - if LMC is below LAC, it pulls the average cost down, and if above, it pushes it up.
• Understanding why the LMC cuts from below is important - as long as adding one more unit costs less than the average so far, the LAC will decline. Once the cost of additional units grows and surpasses the average, the LAC starts to rise.

By studying these curves, firms can strategically adjust production levels to maintain efficiency and minimize costs over the long run, ensuring sustainable growth and profitability.