Derivation Of Long Run Marginal Cost Curve (LRMC) | Ecoholics

Summary

The video provides a detailed explanation of how to derive the long-run marginal cost (LRMC) curve, a topic frequently seen in competitive exams. It begins by defining the LRMC, describing how it shows the additional cost incurred when one more unit of output is produced in the long run. The video explains the relationship between short-run and long-run cost curves, highlighting their U-shaped nature and the impact of economies of scale. Finally, the derivation of the LRMC curve is illustrated graphically, emphasizing its flatter shape compared to short-term marginal cost curves.

Takeaways

• 📉 The long-run marginal cost (LRMC) curve shows the additional cost incurred when one more unit of output is produced in the long run.
• 🧮 Mathematically, LRMC is the change in total cost divided by the change in quantity, or the first derivative of total cost with respect to quantity.
• 📊 The LRMC curve is derived from short-run marginal cost (SRMC) curves, but it tends to be flatter.
• ⬇️ Short-run average cost (SAC) curves are U-shaped due to initially decreasing costs (increasing returns) and later increasing costs.
• 🏭 In the long run, firms can choose between operating on one plant’s increasing cost portion or adding more plants, providing flexibility with inputs.
• 📈 The long-run average cost (LRAC) curve is derived by connecting the minimum points of SAC curves from different plants.
• 🔄 The LRAC curve is often referred to as the 'planning curve' as it accounts for optimal plant size based on economies and diseconomies of scale.
• ⚠️ The SRMC curve intersects the SAC curve at the SAC's minimum point.
• ✏️ The long-run marginal cost curve is derived by connecting the points of tangency between SRMC and LRMC curves at different output levels.
• 🛠️ The LRMC curve also follows a U-shape but is generally flatter than the short-run cost curves.

Q & A

What is the long-run marginal cost (LRMC) curve?

The long-run marginal cost curve shows the additional cost incurred when producing one more unit of output in the long run. It is mathematically defined as the change in total cost divided by the change in quantity produced.

How is the long-run marginal cost curve mathematically represented?

It is represented as the first derivative of total cost with respect to quantity produced, i.e., ΔTotal Cost / ΔQuantity.

What is the relationship between short-run and long-run marginal cost curves?

The long-run marginal cost curve is derived from the short-run marginal cost curves. It tends to be flatter because firms have more flexibility with input factors in the long run.

What is the shape of the long-run marginal cost curve?

The long-run marginal cost curve is typically U-shaped, reflecting economies and diseconomies of scale.

How do short-run average cost curves influence the long-run average cost curve?

The long-run average cost curve is derived from the points of tangency between different short-run average cost curves, representing different plant sizes or production scales.

What is the importance of the tangency points between the short-run average cost and long-run average cost curves?

The tangency points determine the optimal production levels where the firm minimizes costs for each output level. These points are used to construct the long-run average cost curve.

Why does the long-run marginal cost curve tend to be flatter than short-run marginal cost curves?

In the long run, firms have more flexibility in adjusting their inputs, such as deciding whether to produce with one or multiple plants. This flexibility leads to a flatter cost curve as the firm can minimize costs more efficiently.

How is the U-shape of the short-run average cost curve explained?

The U-shape of the short-run average cost curve is due to increasing returns to scale at low production levels, followed by decreasing returns as production increases, which eventually raises costs.

What role do economies of scale play in the long-run cost curves?

Economies of scale reduce the average cost of production as output increases, leading to a downward-sloping portion of the long-run average cost curve. Diseconomies of scale eventually increase the cost, resulting in the U-shape.

How are short-run marginal cost curves related to short-run average cost curves?

Short-run marginal cost curves intersect the short-run average cost curves at their minimum point. This relationship helps identify the optimal production levels in the short run.

Outlines

00:00

📉 Understanding the Long-Run Marginal Cost Curve

The video begins with an introduction to the long-run marginal cost curve, highlighting its importance in competitive exams. It explains that the long-run marginal cost curve shows the additional cost incurred when one more unit of output is produced in the long run. The concept is mathematically denoted by the change in total cost divided by the change in quantity, or the first-order derivative of total cost concerning quantity. The video also notes that the long-run marginal cost curve is typically flatter than the short-run cost curves, reflecting greater flexibility in long-run production decisions.

05:01

📊 Deriving Long-Run Average Cost Curves

This section focuses on the derivation of long-run average cost curves, which are necessary for understanding long-run marginal cost curves. The short-run average cost curves follow a U-shape due to varying returns when more units of output are produced. The firm decides on single- or multi-plant production, which adds flexibility to the long-run production process. The long-run average cost curve is derived from the tangency points of the short-run cost curves, representing optimal production at different scales. These points are connected to form the long-run average cost curve.

10:03

📈 Exploring the Long-Run Marginal Cost Curve

In this paragraph, the long-run marginal cost curve is examined using the previously derived long-run average cost curves. The short-run marginal cost curves are introduced, intersecting the short-run average cost curves at their minimum points. The video describes how the marginal cost points for different plants are marked on the graph, and these points are connected to form the long-run marginal cost curve. This curve also follows a U-shape but is flatter than the short-run curves, reflecting lower marginal costs over a longer period and larger output.

🎯 Conclusion and Final Thoughts

The final paragraph wraps up the discussion by reviewing the main concepts covered, including the derivation of the long-run marginal cost curve and its graphical representation. The video emphasizes that this topic, while important for exams, is relatively simple to understand. The speaker encourages viewers to subscribe, like, and share the content, and invites feedback on future topics they would like to learn about.

Keywords

💡Long-run marginal cost curve

The long-run marginal cost curve (LRMC) represents the additional cost incurred when producing one more unit of output in the long run. It is derived by taking the derivative of the total cost with respect to output quantity. This curve is typically U-shaped and flatter than the short-run marginal cost curve, indicating more flexibility in production and costs over time.

💡Long-run average cost curve

The long-run average cost curve (LRAC) shows the lowest possible cost at which a firm can produce any given level of output in the long run when all inputs are variable. It is derived from the points of tangency of short-run average cost curves and represents a 'planning curve' for firms deciding between different production scales. This curve is also U-shaped due to economies and diseconomies of scale.

💡Short-run marginal cost curve

The short-run marginal cost curve (SRMC) represents the additional cost of producing one more unit of output when some production inputs are fixed. This curve intersects the short-run average cost curve at its minimum point and is U-shaped due to initially decreasing, then increasing marginal costs as production expands.

💡Short-run average cost curve

The short-run average cost curve (SRAC) represents the average cost per unit of output in the short run when at least one factor of production is fixed. Like the marginal cost curve, it is U-shaped due to increasing and then diminishing returns to scale. Firms use multiple SRAC curves to determine the optimal level of output for different scales of production.

💡Economies of scale

Economies of scale refer to the cost advantages that a firm experiences as it increases output, leading to a lower average cost per unit. In the video, this concept explains why the long-run average cost curve initially decreases. As production scales up, firms become more efficient, reducing per-unit costs.

💡Diseconomies of scale

Diseconomies of scale occur when a firm becomes less efficient as it grows, leading to an increase in the average cost per unit. This concept explains the upward slope of the long-run average cost curve after a certain point, as managing larger operations introduces inefficiencies.

💡Minimum efficient scale

The minimum efficient scale (MES) is the lowest level of output at which a firm can achieve the minimum long-run average cost. In the video, it is represented as the point where the firm operates at optimal efficiency, utilizing the minimum value of the short-run average cost curve.

💡Tangency points

Tangency points refer to the points where the short-run average cost curves touch the long-run average cost curve. These points are crucial for deriving the long-run cost curves, as they represent the optimal level of output in the short run for different scales of production.

💡Multi-plant production

Multi-plant production refers to the decision by a firm to operate multiple production facilities, each with its own short-run cost curves. The video explains that firms can open new plants to avoid operating on the increasing-cost portion of the short-run average cost curve, influencing the shape of the long-run cost curves.

💡Planning curve

The planning curve, also known as the long-run average cost curve, represents the firm's long-term production decisions. It envelops all short-run cost curves and shows the lowest cost at which a firm can produce different output levels, serving as a guide for choosing the optimal scale of production.

Highlights

Introduction to long-run marginal cost curve derivation and its relevance in competitive exams.

Definition of the long-run marginal cost curve: Additional cost incurred when producing one more unit in the long run.

Explanation of the mathematical formula for long-run marginal cost as the change in total cost divided by the change in quantity.

Visualization of the long-run marginal cost curve as a U-shaped diagram, derived from short-term marginal cost curves.

Short-run average cost curves exhibit a U-shape due to increasing and decreasing returns to the factors of production.

Discussion of the flexibility of factors in the long run, allowing firms to choose between single or multi-plant production.

Introduction of the long-run average cost curve, derived from points of tangency between short-run average cost curves.

Explanation of how the long-run average cost curve is formed by selecting the optimal production points from various short-run curves.

Long-run average cost curve, also known as the planning curve, reflects the economies and diseconomies of scale.

Description of the process to derive the long-run marginal cost curve using short-run marginal cost curves and points of tangency.

The importance of marking minimum points where short-run marginal costs intersect with average cost curves.

Visualization of multiple short-run average cost and marginal cost curves, leading to the long-run marginal cost curve.

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

Key conclusion: The long-run marginal cost curve follows a U-shape but remains flatter compared to short-run curves.

Final remarks on the ease of understanding the topic and the encouragement to engage with the content further.

Transcripts

[Music]

hello welcome to ecoholics let us today

derive the long run marginal cost curve

the longer an average cost curve is

quite popularly asked but in the recent

years we've also seen the long run

marginal cost curve and its derivation

being asked in competitive exams let's

solve the same

firstly what is a long-run marginal cost

curve

the long-run marginal cost curve shows

the additional cost incurred in the long

run when one more unit of output is

reduced

mathematically we can denote this as the

change in the total cost caused by

one more unit of output so delta total

cost in the long run divided by

change in quantity or we can just say

the first order derivation of total cost

with respect to quantity the same can be

shown

through a simple u-shaped diagram

this

long-run marginal cost curve is derived

from the short-term marginal cost curves

it tends to be flatter than the

short-term

cost curves

let us now

go ahead and derive the same

first of all what we require are the

short-term average cost curves

the short and average cost curves follow

a traditional

u-shaped curve

this u-shaped curve tells us that there

are

different returns to the factor that is

being utilized or different returns when

one more unit of output is being

produced

initially

the average cost is decreasing due to

increasing returns

and

a minimum value is now reached but due

to increasing costs

we see

the average cost curve moving upward

the short-term average cost curves the

short-term marginal cost curves all all

the cost curves they will be following

this u-shaped diagram

but in the long run the factors become

much more flexible the firms can decide

whether

they can they want to produce the output

in only one plant or they want to go for

multi-plant production

so the long-run cost curves are always

going to be

taken over a longer quantity which is

being produced and also

a higher flexibility with respect to

inputs which will be utilized for the

same

let us see how we derive the long run

average cost curves because they are

essential whenever we have to derive the

long run marginal cost curves

in the short run the first plant of a

firm will follow the u shape we just

discussed

now beyond this

output

the firm can decide whether it wants to

operate on the increasing cost portion

of the cost curve or it wants to operate

one more plant

in case it goes for operation or one

more plant we will draw a new short run

average cost curve which will be marked

as acc2

and here we have sac1

similarly it can go ahead and make more

plants

the long run average cost curve is going

to be derived at the points of tangency

or rather we can say

that if the firm has decided to go ahead

and operate at the decreasing cost

portion in the short run of the first

plant

and maybe at the minimum portion of this

plant

showing sac2

and at the increasing portion of these

two other graphs

we join these points of

production

these will tell us what will be the long

run average cost curve

joining the same

we get this

so this is the long run

average cost curve

which is inclusive of all the points of

production

in the short run

and we have taken all the short run

average cost values which were utilized

by the plant

so now

we know that the long run average cost

curve

is

found out

by the firm when it decides whether it

goes for single or multiple plants

therefore this is also known as the

planning curve it is the curve which is

enclosing or the sustaination to all the

short-term cost curves

and

the tangency is according to the optimal

plant size determined by

economies of scale and diseconomies of

scale now let us move to long run

marginal cost curve

for finding the long run marginal cost

curve we will be relying on our old long

run average cost curves

here we will find the point of tangency

of short-run average cost and long-run

average cost and mark the output levels

from here

at the given output we will mark

the value at the short run marginal cost

curve so please be careful

in the short run

if we do have a

average cost curve we will also have the

marginal cost curve which will be

intersecting the average cost at the

very minimum and we are talking about

this short-term marginal cost curve

so we will mark the quantity

value here and the value of the

at the value of shorthand marginal cost

we will be pointing we will be

marking all these points and then

joining all these points to derive the

long run marginal cost curve so here we

have the

theoretical explanation let us just draw

the same again on a graph

so here we have the first plant

having short-term average cost one

as its

curve

and here we have sac2

the average cost curve of the second

plant

and we have

scc3

say the average cost of the third plant

in the short run

here

say the optimal production

in the first plant is at the decreasing

cost portion

at the second plant at the optimal level

or the minimum efficient scale or the

minimum value of short-term average

costs two

and in the third plant

somewhere on the increasing cost

portions

joining these three points

of

short run

equilibriums

we get the long run

average cost curve

at this quantity

we will be making a small marking so we

have x1

x2

and finally

x3

now

we will incorporate the short-term

marginal cost curves

please remember

shortened marginal costs or any marginal

cost will always intersect average cost

curves at their minimum point

so for simplicity we will be marking the

minimum points

and we will give it we will give a

u-shape to the short-term marginal cost

curves

so here we have smc one depicting the

short run marshall cost of the first

plant

we have smc 2

which is depicting the short-run

marginal cost of the second plant

and finally

smc 3

which is depicting the shortened

marginal cost of the third plant

now we will be utilizing the

portions or the quantities that we had

marked

the quantity marked

x1 was being produced at the marginal

cost of this value

so we will be marking

it with green here

and here

at the quantity x 2 the value of

short-term marginal cost is again at

this point the convergence of the

two lines

and here we will extend the quantity

upward

and

to produce

x3 the shortened marginal cost was

equivalent to this entire value this

entire value

on the graph

now

to finally derive the long run marginal

cost we will simply join all these three

points

and here we have

the long run marginal cost curve

which is again following a u-shaped

but it is much flatter than the

short-term marginal cost curves so again

we covered a very important topic and we

can realize that this is a very easy

topic

so please

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