Deriving the Long Run Marginal Cost Curve

econhelp

14 Sept 202208:11

Summary

TLDRThis video explains the derivation of the long-run marginal cost (LRMC) curve, starting with a review of the long-run average cost (LRAC) curve. The LRAC is the lower envelope of a firm's short-run average cost curves, where capital is fixed in the short run but flexible in the long run. The LRMC is derived by tracing marginal costs at different quantities, showing how it is flatter than short-run marginal cost curves due to the firm’s ability to adjust inputs. The LRMC also intersects the LRAC at its minimum, with key insights into their relationship.

Takeaways

📉 The long-run marginal cost curve is derived from understanding the long-run average cost curve.
📊 The long-run average cost curve is the lower envelope of the firm’s short-run average cost curves.
🔄 In the short run, the firm faces a fixed input, typically capital, which leads to different short-run average cost curves depending on the level of capital.
🏗️ In the long run, the firm can choose any level of capital, selecting the one that minimizes production costs for each output level.
📐 The long-run marginal cost for each quantity is derived from the corresponding short-run marginal cost curve.
🔗 The long-run marginal cost curve is created by connecting points from short-run marginal cost curves that correspond to the firm’s chosen level of capital for each output quantity.
📍 The marginal cost always intersects the minimum point of the average cost curves, both in the short run and the long run.
🔄 The long-run marginal cost curve is flatter than the short-run marginal cost curves because the firm has more flexibility in adjusting inputs over time.
📉 The long-run marginal cost curve passes through the minimum point of the long-run average cost curve, following a logic similar to short-run curves.
🔗 The relationship between long-run curves and total cost curves can be better understood with additional diagrams, as mentioned in the video.

Q & A

What is the first step in understanding the derivation of the long-run marginal cost (LRMC) curve?
-The first step is to understand the derivation of the long-run average cost (LRAC) curve, which is shown as the lower envelope of the firm's short-run average cost (SRAC) curves.
How is the long-run average cost curve derived?
-The long-run average cost curve is derived by tracing the lower envelope of all the firm's short-run average cost curves, which correspond to different levels of capital. In the long run, the firm can choose the level of capital that minimizes production costs for each quantity.
What does each short-run average cost curve represent?
-Each short-run average cost curve represents the firm’s average costs when a particular input, such as capital, is fixed at a certain level. These curves differ depending on the level of fixed capital.
Why does the long-run average cost curve appear smoother than the short-run average cost curves?
-The long-run average cost curve is smoother because it is the lower envelope of many short-run average cost curves, each representing different levels of capital. Since the firm can adjust its capital in the long run, this creates a smoother curve.
How is the long-run marginal cost (LRMC) curve constructed?
-The LRMC curve is constructed by tracing the long-run marginal cost for each quantity produced. For each level of output, the corresponding short-run marginal cost (SRMC) curve is used, and the long-run marginal cost is read off from this curve.
At what point are the long-run marginal cost and long-run average cost equal?
-The long-run marginal cost and long-run average cost are equal at the minimum point of the long-run average cost curve. This occurs because marginal cost always intersects the average cost at its minimum.
Why is the long-run marginal cost curve flatter than the short-run marginal cost curve?
-The LRMC curve is flatter than the SRMC curves because, in the long run, the firm has the flexibility to adjust all inputs, such as labor and capital. This allows the firm to choose the most cost-efficient combination of inputs, leading to slower increases in marginal cost compared to the short-run, where some inputs are fixed.
What happens to the relationship between marginal cost and average cost when average costs are decreasing?
-When average costs are decreasing, marginal cost must be lower than average costs. This relationship holds for both short-run and long-run cost curves.
What is the significance of the tangency point between the long-run average cost curve and short-run average cost curve?
-The tangency point indicates the optimal level of capital for a given level of output. At this point, both the short-run and long-run average cost curves are at their minimums, and the marginal cost equals the average cost.
Why is the relationship between long-run cost curves and total cost curves important to understand?
-Understanding the relationship between long-run cost curves and total cost curves helps to see how costs behave as output expands. This can offer insights into cost structures, economies of scale, and production efficiency in the long run.

Outlines

00:00

📈 Derivation of Long Run Average Cost Curve

The first paragraph explains the process of deriving the long run average cost curve. It starts by emphasizing the need to understand the firm's short run average cost curves, which are fixed at different levels of capital. The long run average cost curve is described as the lower envelope of these short run curves, indicating the minimum cost of production for different quantities. The process involves selecting the level of capital that minimizes costs for each quantity produced. The paragraph also mentions that in the long run, the firm can vary all inputs, leading to a smoother long run average cost curve. A longer video on this topic is offered for further details.

05:00

📊 Constructing the Long Run Marginal Cost Curve

The second paragraph focuses on constructing the long run marginal cost curve. It explains that, similar to the long run average cost curve, the marginal cost curve is derived by tracing the lowest points of the short run marginal cost curves at each quantity. The video script illustrates this by showing how to find the marginal cost at specific quantities, such as q star and q prime. It also discusses two key features of the long run marginal cost curve: its flatter shape compared to the short run curves due to the flexibility in input combinations, and its intersection with the minimum point of the long run average cost curve, reflecting the economic principle that marginal cost equals average cost at the minimum point of average cost.

Mindmap

Keywords

💡Long Run Marginal Cost (LRMC)

Long run marginal cost refers to the additional cost incurred by producing one more unit of output in the long run when all inputs are variable. In the video, LRMC is derived by examining how firms choose optimal input levels over different quantities of output, with the ability to adjust factors like capital. The speaker emphasizes that LRMC is flatter than short run marginal cost curves due to the flexibility of input choices in the long run.

💡Long Run Average Cost (LRAC)

Long run average cost represents the average cost of production when all inputs are adjustable. It is derived as the 'lower envelope' of various short run average cost curves, showing the least cost at which any given level of output can be produced in the long run. The video explains that LRAC smooths out as firms adjust inputs like capital over time to minimize costs.

💡Short Run Average Cost (SRAC)

Short run average cost is the average cost of production when at least one input, such as capital, is fixed. The video illustrates this concept by showing how SRAC curves vary depending on the level of capital employed. Multiple SRAC curves exist, corresponding to different fixed levels of capital in the short run.

💡Short Run Marginal Cost (SRMC)

Short run marginal cost is the cost of producing one more unit of output when some inputs are fixed, typically capital. The video contrasts SRMC with LRMC, showing that SRMC curves are steeper due to constraints on input flexibility. These constraints cause SRMC to increase faster than LRMC.

💡Capital

Capital refers to the fixed input in the short run, such as machinery or buildings, that cannot be adjusted immediately. The video uses capital as the primary example of a fixed input that influences both SRAC and SRMC. In the long run, capital becomes a variable input, which helps firms minimize costs and adjust their production levels more efficiently.

💡Lower Envelope

The 'lower envelope' describes how the long run average cost curve is derived as the minimum points of multiple short run average cost curves. The video shows that this lower envelope represents the least costly way for firms to produce each output level, assuming they can adjust inputs like capital in the long run.

💡Cost Minimization

Cost minimization is the process by which firms choose the combination of inputs that allows them to produce a given level of output at the lowest possible cost. The video explains that in the long run, firms can vary both capital and labor to minimize costs, which is reflected in the shape of the LRAC curve.

💡Tangent

In the video, 'tangent' refers to the point where the short run average cost curve touches the long run average cost curve at its minimum. At this point, the short run and long run marginal cost curves are equal, and it represents the most efficient level of production in both the short run and the long run.

💡Marginal Cost and Average Cost Intersection

This concept refers to the fact that marginal cost intersects the average cost curve at its minimum point. In the video, the speaker explains that this is true for both short run and long run curves, meaning that when average costs are minimized, marginal costs must be equal to average costs.

💡Input Flexibility

Input flexibility refers to a firm's ability to adjust its use of inputs, such as capital and labor, over time. The video contrasts input flexibility in the long run, where firms can freely adjust all inputs, with the short run, where some inputs are fixed. This flexibility is key to understanding why long run cost curves tend to be flatter than short run curves.

Highlights

Introduction to deriving the long-run marginal cost curve and its connection with the long-run average cost curve.

Review of the derivation of the long-run average cost curve as the lower envelope of the firm’s short-run average cost curves.

Explanation of how a firm can choose the optimal level of capital in the long run to minimize production costs.

Clarification that the long-run average cost curve is smoother due to the many short-run average cost curves being considered.

Derivation of the long-run marginal cost curve by analyzing short-run marginal cost curves at different quantities.

Key concept: at the minimum point of the long-run average cost curve, the marginal cost equals the average cost.

Visual representation of long-run marginal cost curve points based on short-run marginal cost curves.

The long-run marginal cost curve is flatter than the short-run marginal cost curves due to flexibility in input choices in the long run.

The long-run marginal cost curve passes through the minimum of the long-run average cost curve, as average costs decrease when marginal cost is lower.

The marginal cost curve's behavior explains whether average costs are increasing or decreasing.

Illustration of how firms in the short run are constrained by fixed inputs, but not in the long run.

Emphasis on how firms can adjust capital and labor inputs in the long run to find the most cost-efficient production levels.

The long-run marginal cost curve increases more gradually compared to the short-run marginal cost curve.

Summary of the importance of understanding the relationship between marginal cost and average cost curves for both short-run and long-run decisions.

Mention of additional videos that provide more detailed explanations and alternative diagrams to further understand the relationships between cost curves.