in this video I'm going to demonstrate
how to derive the firm's seven short-run
cost functions from a cop Douglas
production function take the
cobb-douglas production function of Q
equals L to the 1/2 times K to the 1/2
power L is units of labor k units of
capital flips units of capital q is the
firm's output quantity of output a few
things that we're going to need in this
problem since we're driving short-run
cost functions capital is going to be
fixed at some level so let's let k equal
25 a few other things that we'll need
our firm's wage with the firm pays each
worker call that W for sure say that
equals 100 dollars and then we need the
firm's rental rate of capital firms
price of capital
what are represent that and we'll set
the rental rate of capital equal to $500
so again just to recap we have a cop
Douglas production function and we want
to derive the firm's seven short-run
cost functions the first step is to take
this production function and solve for
out
so I'm going to substitute our level of
capital 25 in 4k and then take the
square root of that just rewriting and
now solve for L look over here
dividing both sides through by five and
then what we need to do now is square
both sides and square both sides get rid
of the square root of L and squaring
five here in the denominator gives us 25
so this is a key result for us and I'll
show you what to do with this in a
second okay moving down here at ad we're
going to start with the firm's total
cost so total cost equals variable cost
or some call it total variable cost plus
fixed cost and again some books or
professors may refer to that as total
fixed cost so total cost is the sum of
the firm's variable cost of production
plus the firm's fixed cost of production
in this example our only variable cost
is going to be labor so variable cost is
going to be W the wage times labor
number of units of labor and our fixed
costs are only fixed cost is going to be
capital the price of capital R the
rental rate is R times the units of
capital so what we're going to do now is
substitute in what we know for these
values W is a hundred we let W equal 100
L we just solve is Q squared divided by
25
the price of capital is 500 and we're
multiplying that by K and we're dealing
with 25 units of capital so let's just
simplify that slightly total cost equals
4 Q squared under divided by 25 gives us
4 plus 500 times 25 12,500 so this is
one of the firm's seven cost functions
of firms total cost let me get a new
sheet here let me just rewrite so if the
firm's total cost is 4 Q squared plus
12,500 we can now easily solve the other
six cost functions for the firm
the firm's variable cost it's going to
be this component the component that
depends on the quantity of output for Q
squared
the firm's fixed cost is going to be the
constant 12,500 this is the amount the
firm has to pay over some time period
regardless if they produce zero units
two units or thousand units the next
thing we need is the firm's average
total cost average total cost or the per
unit cost of production equals the
firm's total cost divided by quantity
so we're just going to take the firm's
total cost function and divide it
through by Q if we do that we're going
to get four Q plus twelve thousand five
hundred divided by Q so this is the
firm's average total cost now let's get
the firm's average variable cost average
variable cost is variable cost divided
by Q or 4q squared divided by Q equals
4q and the firm's average fixed cost is
fixed cost divided by Q which is twelve
thousand five hundred divided by Q one
thing you'll note here you may recall
that the average total cost is average
variable cost plus average fixed cost so
if you look at our Navarrete total cost
equation notice this 4q well that's
everyda variable cost and this twelve
thousand divided by twelve thousand five
hundred divided by q that's average
fixed cost see it down here alright we
need one more cost and that's the firm's
marginal cost marginal cost is a
derivative concept we're going to take
the derivative of the total cost
equation with respect to output and if
we do that we're going to get a Q the
derivative of for Q squared is just a Q
the derivative of a constant is zero so
our marginal cost again is just a
derivative of this top equation and we
get a Q it's also as a little side here
marginal cost is also nothing more than
the derivative of the variable cost
equation with respect to output since
fixed costs don't change the derivative
of total cost is going to with respect
help but it's going to be the same thing
as a derivative of variable cost with
respect to output so if we take the
derivative of the variable cost equation
for Q squared once again we get back
eight Q and that is seven of the firm's
all seven of the firm's short-run cost
functions hope you found this beneficial