Derivation Of Long Run Marginal Cost Curve (LRMC) | Ecoholics
Summary
TLDRThe 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
📉 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.
📊 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.
📈 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.
Mindmap
Keywords
💡Long-run marginal cost curve
💡Long-run average cost curve
💡Short-run marginal cost curve
💡Short-run average cost curve
💡Economies of scale
💡Diseconomies of scale
💡Minimum efficient scale
💡Tangency points
💡Multi-plant production
💡Planning curve
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
subscribe to our channel like our
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