Deriving the Long Run Marginal Cost Curve
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
📈 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.
📊 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 Average Cost (LRAC)
💡Short Run Average Cost (SRAC)
💡Short Run Marginal Cost (SRMC)
💡Capital
💡Lower Envelope
💡Cost Minimization
💡Tangent
💡Marginal Cost and Average Cost Intersection
💡Input Flexibility
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.
Transcripts
hi everyone in this video i'm going to
discuss deriving our long run marginal
cost curve
and really in order to understand the
derivation of our long run marginal cost
curve we first need to understand the
derivation of our long run average cost
curve which i've drawn here
as the lower envelope of the firm's
short run average cost curves
because we do need this background the
plan in this video is to firstly briefly
review the derivation of our long run
average cost curve
i'm then going to go on to find our
long-run marginal cost curve and lastly
in the third part of the video i'll
discuss two features of the long run
marginal cost curve which are good to
know
if you're already comfortable with
deriving the long run average cost curve
you can just skip straight to the second
part of the video all of these parts are
time stamped in the description
okay so just really quickly
in order to derive our long run average
cost curve we first have to recognize
that in the short run the firm faces a
fixed input to production so usually we
say that capital is fixed
and this really means that when we draw
a single short run average cost curve
say something like i have here
well this curve shows the firm's average
costs when capital is fixed at some
level
of course there are lots of different
levels of capital that the firm could
hold in the short run however and it
follows that there are many different
possible short run average cost curves
each of which corresponds to a different
level of capital so for illustration
i've just drawn
five short run average cost curves here
and this corresponds to five different
levels of capital now in the long run
the firm can choose whichever level of
capital that they wish
and actually the determining factor
in this decision is the level of capital
that minimizes the cost of production
for the firm
so to find the long run average cost
associated with producing any amount so
let's just say q star
what we're going to do is we're going to
trace a line up
and the first curve that we hit well
that corresponds to the cheapest
possible
way available to the firm of producing q
star
and so that level here will be the long
run average cost for producing q star
and the level of capital that
corresponds to that short run average
cost curve will be the level of capital
that the firm will choose in the long
run if it wants to make q star units
if we do the same exercise for all of
the possible quantities that the firm
could produce we essentially trace out
the lower parts of our short run average
cost curves hence the idea of the long
run average cost curve as being the
lower envelope of our short run average
cost curves
lastly once we recognize that a firm can
hold many many different levels of
capital and each of those levels of
capital would be associated with a
unique short run average cost curve so
something like what i have here and even
more short run average cost curves then
once we take the lower envelope here we
get a much smoother long run average
cost curve which is hopefully familiar
to you from your textbooks
so that was a very short summary of
finding our long run average cost curve
i do have a longer video on deriving our
long run average cost curve that i'll
link to below just in case you need that
extra detail
in order to find our long run marginal
cost curve i'm going to do a very
similar thing except i'm going to trace
out the long run marginal cost for each
quantity
so let's take a point let's look at q
star again
and we know from the derivation of our
long run average cost curve that there
is an associated short run average cost
curve
that is set at the level of capital that
the firm will choose in the long run if
it wants to produce q star
but this time i'm also going to add in
the short run marginal cost curve that's
associated with that short run average
cost curve and similar to the short run
average cost curve because this short
run marginal cost curve is constructed
with capital set at the level that
minimizes the cost of producing q star
so that long run level of capital that
the firm will choose if the firm
produces q star in the long run well we
can read off from that marginal cost
curve the long run marginal cost of
producing q star so it will be right at
this level here and that will be just
one point on our long run marginal cost
curve and i'll indicate that uh point
with a red mark
so we can take another point maybe q
star star just here and this will get
messy so i've taken away those previous
curves for q star
but what we'll do is we'll add in our
short run average and our short run
marginal cost curves and again we can
read that long run marginal cost just
off that short run marginal cost curve
just like that and this is how we
construct the long run marginal cost
curve
now there will be a point right here at
the minimum of our long run average cost
curve let's just call that q prime
and at this point the corresponding
short run average cost curve will be at
a minimum as well so the two curves will
actually be a tangent to one another
and at this point the marginal cost
curves will be actually equal to the
average cost curves and this is because
marginal cost always intersects the
minimum of our average cost and that's
true for both the short run and the long
run curves
and so at this level here well our long
run marginal cost for q prime is the
same as our long run average cost for q
prime let's just do one more point let's
think about q prime prime and actually
on this side of our long run average
cost curve so the right hand side of the
curve our marginal cost i can hope you
can see is above our average cost so
just up here like this and so i hope you
can see that there
all right so in order to get our long
run marginal cost curve we can just
connect all of these points and i get
something like
these red this red line here
of course i've only demonstrated this by
looking at four points and similar to
our long run average cost curve you can
imagine that if we find the long run
marginal cost associated with every
possible quantity that the firm could
produce
we would get a smoother curve so
something like this
so that's the derivation of our long run
marginal cost now there are two features
of our long run marginal cost curve that
i would like to point out
the first feature is that our long run
marginal cost curve is really much
flatter than our short run marginal cost
curves which are pretty steep like this
and this is because in the short run the
firm is constrained in its use of
capital
in the long run if the firm wants to
produce one more unit of output
well the firm is free to vary the amount
of capital or the amount of labor or any
of its other inputs that they that they
use and so they can choose the cheapest
combination of inputs in order to get
that additional quantity
in the short run the firm doesn't have
this freedom it's constrained in its use
of capital and as a result the long run
marginal cost curve increases at a
slower rate in comparison to the short
run marginal cost curve and is thus
flatter
the second feature is one that i
mentioned before and that is that the
long run marginal cost curve will go
through the minimum of our long run
average cost curve
and to explain this we can refer to our
usual explanation that we invoke in the
case of our short run cost curves and
that is well if average costs are
decreasing then marginal cost must be
lower than average costs if average
costs are increasing then marginal cost
must be above average costs i do have
another video on that logic and i'll
link to it below
the video is on the short run case but
the logic is the same in both the short
and the long run so it will still be
useful for you if you need it
all right that's it there is one other
video on this stuff that i did want to
do which includes a diagram that links
the relationship of our long run curves
to
the firm's total cost curve so when i
get around to it i'll link to it below
it's an alternative way of seeing all
these relationships um and so it's
pretty neat
all right i hope that that did help if
it did please like and subscribe thanks
to my subscribers so far and i hope you
guys are having a lovely day or night
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