Suppose we're given the following
Cobb-Douglas production function, P of L comma K,
where L is the units of labor,
K is units of capital,
and P of L comma is total units that can be produced
with this labor/capital combination.
Suppose each unit of labor costs $900,
and each unit of capital costs $1,800.
Also suppose a total of 1 million, 80 thousand dollars
is available to be invested in capital and labor combined.
How many units of labor and capital should be purchased
to maximize production subject to the budgetary constraint,
and also what is the maximum number of units of production
under the given budgetary conditions?
We want to maximize P of L comma K,
under the given constraint.
Let's write the constraint as an equation.
Again, L is the units of labor,
and K is the units of capital.
Labor costs $900 a unit,
and capital costs $1,800 per unit,
which means 900L
plus 1,800K
must equal the total budget
of 1 million, 80 thousand dollars.
Let's go ahead and simplify this by dividing through by 100.
We can use the equivalent constraint,
nine L plus 18K
equals 10,800,
and now we'll use the method of Lagrange Multipliers
to maximize P under this constraint.
In general, if you use the method of Lagrange Multipliers,
if we have a function f of x comma y,
function f of x comma y
is going to be maximized or minimized
under the constraint g,
when the gradient of f is equal to some multiple,
lamba times the gradient of g,
which gives us these first two equations,
and the third equation is our constraint,
g of x comma y equals zero.
In our case, the function f is actually P of L comma K,
and then for our constraint, g of x comma y,
we have g of L comma K,
and because our constraint is nine L plus 18K equals 10,800,
to have our function equal to zero,
we'll subtract 10,800 on both sides,
which would give us g of L comma K
equals nine L plus 18K
minus 10,800 equals zero,
which means our system of equations are going to be
the partial of P with respect to L
equals lambda times the partial of g with respect to L,
and we'll have the partial of P with respect to K
equals lambda times the partial of G with respect to K,
and of course we'll have our constraint,
g of L comma K
equals zero.
Let's go ahead and find our partial derivatives.
To find the partial of P with respect to L,
we'll treat K as a constant,
so we're going to multiply by 0.9.
30 times 0.9 is 27,
then we have L raised to power of 0.9 minus one,
that's negative 0.1,
times K to the 0.1
equals lambda times the partial of g with respect to L,
so we'll treat K as a constant,
so the partial with respect to L would just be nine,
so times nine.
Now, for our second equation,
we have the partial P with respect to K,
so we differentiate here with respect to K,
treating L as a constant,
so we'll multiply by 1/10.
30 times 1/10 would be three
L to the 0.9,
K to the 0.1 minus one.
That'd be negative 0.9
equals lambda times the partial of g with respect to K,
so we're treating L as a constant,
so the partial with respect to K would just be eighteen.
Of course G of L comma K equals zero
is going to be nine L
plus 18K
minus 10,800 equals zero.
Now we have to solve this as a system of equations,
so let's go ahead and solve
these first two equations for lambda.
For the first equation, we would divide both sides by nine,
so we'd have lambda equals
27 L to the negative 0.1,
K to the positive 0.1 divided by nine,
and here we'd have lambda equals
three L to the 0.9 K to negative 0.9
divided by 18.
Now, because both of these expressions are equal to lambda,
we can set them equal to each other and solve
to determine the relationship between L and K.
When we do this, though, let's make the exponents positive,
so we'll move L to the negative 0.1 down to the denominator,
as well as K to negative 0.9,
so this would give us the equation 27
K to the 0.1
over, we still have a nine,
and then we'd have L to the positive 0.1 equals,
here we'd have three L to the 0.9
divided by 18, and then we'd have K to the positive 0.9.
Again, this is true, because both of these
are equal to lambda.
Now we'll cross multiply to determine
the relationship between L and K,
which means this product must equal this product,
so we'd have 27K to the 0.1
times 18K to the 0.9
must equal nine L to the 0.1
times three L to the 0.9.
27 times 18 is equal to 486,
then K to the 0.1 times K to the 0.9
would be equal to just K.
Remember, we'd be adding the exponents.
0.1 times 0.9 is equal to one,
so we just have K to the first
equals on the right, nine times three is 27,
and we have L to the 0.1 times L to the 0.9,
which would be just L to the first, or just L.
To solve this equation for L,
we'll divide both sides by 27,
which gives us L equals
486K divided by 27, would be equal to 18K.
Now that we know that L is equal to 18K,
what we're going to do
is substitute 18K for L in our constraint,
which will give us one equation and one variable,
which will allow us to find K as well as L.
Let's do this on the next slide.
Again, we know L equals 18K,
and we also know the constraint
is nine L plus 18K
minus 10,800 must equal zero,
so if we substitute 18K for L,
we have nine times 18K
plus 18K
equals 10,800.
Well, nine times 18 is equal to 162,
so we have 162K
plus 18K,
which equals 180K.
Dividing both sides by 180,
we have K equals 60.
Well, now we know K equals 60 and we know L equals 18K,
we know that L equals 18 times 60,
which equals 1,080.
Remember, L equals the units of labor,
and K is equal to the units
of capital.
Going back to our question,
we're asked how many units of labor and capital
should be purchased to maximize production,
subject to the budgetary constraints.
We know the units of labor L is equal to 1,080,
and the units of capital K is equal to 60.
Now, to find the maximum number of units of production
under the given budgetary constraints,
we need to substitute these values in
for L and K into our function P.
Let's go ahead and do this.
We'd have P of 1,080 comma 60,
which would be equal to 30 times
1,080 raised to the 0.9
times 60 raised the power of 0.1.
Going to the calculator, we have 30 times 1,080
raised to the power of 0.9.
Right arrow,
open parenthesis 60, close parenthesis,
raised to the power of 0.1.
Rounding to the nearest unit,
rounding down, we'd have 24,267,
which means, under these constraints,
the maximum production, again,
is 24,267 units.
Before we go, though, let's take a look
at what's happening graphically.
We're going to graph the level curves
for P of L comma K,
as well as the constraint, g of L comma K.
We see the level curves of P and g,
graphed here in blue and green,
and these two vectors here,
the black vector and the red vector,
are the gradient of P and the gradient of g.
Notice how where the level curves are tangent to one another
both the gradient of P and the gradient of g
are multiples of one another,
which is the whole idea behind
the method of Lagrange Multipliers,
so this point here is the point where the production
is maximized under the given constraints.
I hope you found this helpful.