Long Run Costs - Economics Video Lecture | Microeconomics- Interaction between individual buyer-seller

this tutorial is an introduction to cost
theory in the long run it's part one and
a 3 video playlist the next video I will
use numbers and finally on the third
video I will do a calculus proof for you
so let's move on for the first video I
put labor on the x-axis and capital on
the y-axis and I use K for capital and L
for labor the Green Line is called an
ISO cost line which means equal so the
Green Line is equal cost everywhere
along this line the producer has an
equal cost with a mix of capital and
labor total cost is equal to the rate of
capital or the cost of capital times the
amount of capital used plus labor costs
which are the wage rate times the
quantity of labor used and to be clear
I'm talking about the quantity of
capital and the quantity of labor
this is typically written as C for cost
is equal to R the rate of capital x
capital which is k plus w for the wages
times labor so I end up with C is equal
to RK plus WL I'm going to solve this
equation in terms of K which is the same
as Y solving it for y that K right there
so I subtract WL from both sides of the
equation that cancels out so I pull down
RK is equal to C so I have C minus W
times L is equal to RK now when we get
rid of that our move it to the other
side of the equation and I do that by
multiplying 1 over R both sides of the
equation so on the left hand side I have
C divided by R minus WL divided by r
this is equal to those RS cancel out and
this is equal to K and this is the
equation for that line cost is constant
so a lot of times you'll see the line
put over the top of the C that's what
that means frequently it's written as K
is equal to all this instead and let me
just rewrite that part of the equation a
little different so it becomes W over R
times L
now you can see that negative W over R
is the slope of that line so if I let L
equals zero
I'll put zero in for the equation right
there so labor 0 what will capital be so
that cancels out so capital is equal to
C divided by R when labor zero and
that's the y-intercept to that point
right there so at that point right there
labor is 0 and capital the amount of
capital used is the total cost divided
by the cost of capital now if I don't
use any capital I have K is equal to
zero how much labor will be used so K is
0 I'm going to determine that point
right there so I plot in 0 0 is equal to
that equation and I solve for L so I
subtract C / R from both sides of the
equation those two cancel out so I'm
left with negative C divided by R is
equal to negative W divided by R times L
I want to isolate L so I multiply both
sides of the equation by negative R
divided by W
and now those cancel out right there and
that cancels with that are in that are
also the negative signs cancel this
leaves me with C divided by W is equal
to L so when capital is zero that point
right there it's C divided by W it's the
amount of labour used so in capital is
equal to zero the amount of labour used
is C divided by W in other words when
capital is equal to zero the amount of
labour used is total cost divided by the
amount spent on labour and the slope of
that line is equal to negative W divided
by R I can also determine the slope
another way let me show you if I use K 1
capital I will use L 1 labor and at that
point K 2 capital and L 2 labor so I
have K 2 minus K 1 divided by that's a
change in capital by the way divided by
L 2 minus L 1 or the change in labor
this is equal to the change in capital
divided by the change in labor is
remember rise over run this can be
written as now or the slope of the line
is negative change in capital divided by
change in labor this is equal to the
negative W divided by R so those are
equivalent
both slope of that line
keep this in your mind I'll come back to
this now I'm going to move on to a thing
called ISO quant or equal quantity along
this line here the producer has equal
amounts of quantity so as they move
along that line the change in quantity
is zero so we have we write it like that
change in quantity is equal to zero or
change in Q is equal to zero
I'm going to add in other quantity lines
so I have q1 q2 q3 and q1 quantity of
that of that iso quant line is less than
the quantity produced at that ISO a line
of q2 which is less than q3
so as I move from q1 to q3 quantity
increases the amount of quantity
produced on a single iso quant line is
equal to quantity actually is a function
of the amount of labor and amount of
capital used so I have k1 there in l1 if
I reduce the amount of capital used I
need to increase the amount of labor so
if I want to keep quantity constant I
have to increase the amount of labour
used from l1 to l2 and again quantity is
a function of Labor and capital and this
time it's l2 and k2 so these are equal
the amount of quantity is equal for both
those functions or the change in
quantity is 0 the change in capital
times the marginal productivity of
capital marginal product of capital MP
is equal to the change in labour times
the marginal product of labor this means
when I reduce capital if I reduce
capital it has to be offset by that's
how the equation so these remain in
equilibrium or equal to each other if I
multiply both sides of the equation
times one divided by a marginal product
of capital those cancel out so I have
the change in capital is equal to the
change in labour times marginal product
of labor divided by marginal product of
capital now I want to get changing labor
over with change in capital so I
multiply both sides of the equation by 1
divided by changing labor so those
cancel out nicely and I'm left with the
change in capital divided by the change
in labour is equal to marginal product
of labor divided by marginal product of
capital and this is important concept
remember that the slope of the iso cost
line is equal to change in late capital
divided by changing labour also
this is equal to the negative W divided
by R the producer will produce at a
point where all three of these things
are equal the slopes notice the ISO cost
line the green line is the tangent to
the ISO quantity line which is the blue
line this relationship is also known as
the marginal rate of technical
substitution or M R T as marginal rate
of technical substitution up next in the
next video I'm going to do the same idea
but I'll use numbers the same concept as
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