Tutorial 7: Production (cont.)
Production with Two Variable Inputs: Isoquants
In the first part of this Tutorial you viewed an Excel
workbook containing a 3D Surface chart illustrating the
following production function:
Q(L, K) = 1 * L^{0.5} * K^{0.5}
If you haven't done so already, download the workbook
now. [NOTE: The size of this file is larger (~285kb) than
all of the others, with the exception of the Comparative
Statics Quiz, so allow for a longer than normal download
time.]
Click on the Chart tab. When you look at the chart
you notice a line running around this "hill" at
various intervals, representing higher levels of output
(Q) = 20 units/t (at the top of the blue range), 40 units/t
(top of the purple range), and 60 units/t (top of the yellow
range). Each of these lines shows that it is possible to
get a given level of output, say Q = 20 units/t, with various
combinations of L and K. These lines are referred to as
isoquants because they show a constant ('iso') level
of quantity ('quant') resulting from various combinations
of L and K.
If you click on the Data tab and look at the table used
to generate the chart, you will see much the same thing.
Various combinations of L and K have been highlighted that
would generate an output level Q = 10 units/t (see Table
5 below).
Table 5
This reveals a fact of fundamental importance in the theory
of production. It is possible, using the production method
modeled by the CobbDouglas production function, to substitute
less of one input for more of another and still keep output
constant. In the case illustrated by the Data, we can increase
the amount of labor used in the production process from
10 to 20 units/t, and simultaneously drop the amount of
capital used from 10 to 5 units/t with no change in output.
There is more than one way to produce a given level of output!
Diminishing Returns Again…
So what have we got? The Chart (and Data) help
us see that, for a given level of output, when L is increased,
K can be decreased. These lines of constant output are referred
to as isoquants. You can easily modify the chart
type in Excel to view these isoquants in two dimensions.
In the Production.xls workbook, click once on the Chart
tab then, from the Chart menu, choose Chart type.
Select the chart subtype Contour (second row first
chart) and click OK.
The slope of the isoquant reveals how much K may be given
up when one more L is used to produce a given level of Q.
The absolute value of the rate at which L may be substituted
for K is referred to as the Marginal Rate of Technical
Substitution between L and K (MRTS_{LK}). Let's
work with this mathemagically and see what we find...
MRTS is measured as the negative of the slope of
an isoquant (which, of course, is negative). So
MRTS = —DK/DL
> 0
for a given level of Q, where DK
is the "rise" and DL
is the "run". If you look back at the Data in
the Production.xls workbook, you will see that as the amount
of labor used in production increases from 4 to 5 units/t,
the amount of capital that may be "retired", keeping
output constant at 10 units/t, is 5 (from 25 to 20 units/t).
So MRTS is 5 here. But when the amount of labor used in
production increases from 10 to 20 units/t, the amount of
capital that may be "retired", keeping output
constant at 10 units/t, is also 5 (from 10 to 5 units/t).
So MRTS is 0.5 here. So MRTS diminishes as the quantity
of labor hired increases. But why?
Let's see. Normally, when a firm increases the
amount of one input used, keeping the other constant, output
increases. In this example, an increase in L will cause
an increase in Q = MP_{L}*DL.
That is, the additional amount of labor hired, DL, will
raise output by an amount equal to the additional output
that extra labor makes (i.e., its marginal product, MP_{L}).
Similarly, when the firm decreases the amount
of one input used, keeping the other constant, output decreases
by an amount equal to the additional output that input would
have produced. In this example, a decrease in K will
cause a decrease in Q = MP_{K}*DK.
That is, the amount of K laid off, DK, will decrease output
by an amount equal to the marginal product of that capital,
MP_{K.}
If we substitute more labor for less capital along an isoquant,
then the change in output would be zero  the additional
output created by the new labor (MP_{L}*DL)
would be exactly offset by the loss of output resulting
from the decrease in capital (MP_{K}*DK).
Thus:
DQ = 0; MP_{L}*DL
+ MP_{K}*DK = 0
Rearranging terms:
MP_{L}*DL
= —MP_{K}*DK
—DK/DL
= MP_{L}/MP_{K}
\
MRTS = MP_{L}/MP_{K}
So we have found that the rate at which L may be substituted
for K is equal to the ratio of their respective marginal
products. Earlier we found that the marginal rate of technical
substitution declines as we move along the isoquant. How
does knowing how to calculate MRTS answer the question of
why MRTS declines as more L is substituted for K?
Well, in the beginning of this Tutorial we learned that
the marginal product of a single variable input eventually
declines as more of that input is used with all other inputs
held constant. This "law of diminishing returns"
was due to the fact that that variable input, say labor,
had to share the fixed amount of capital that was available.
It is that sharing that promotes diminishing returns. Turning
this around, as less labor is used marginal product rises
because less sharing of capital is required.
So as you move left to right along our isoquant, L increases,
causing MP_{L} to decline, and K increases, causing
MP_{K} to increase. If MRTS = MP_{L}/MP_{K},
then as you move left to right along an isoquant the numerator
decreases (e.g., 5, 4, 3, 2, 1) while the denominator increases
(e.g., 1, 2, 3, 4, 5), resulting in a smaller value for
the ratio (e.g., 5, 2, 1, 0.5, 0.2).
Now it's time to
"do the thing".
Click on the following link
to download the Minimizing
Production Costs Workbook. Work through
Tutorial 7 Questions 2  4. This will help
to improve your understanding of how we model
the production of output using two
variable inputs.
Return here when you have finished.
Need help
downloading the Excel file? 

Now we see that we can model production with two variable
inputs by using a graphical model of isoquants. Isoquants
show various combinations of two inputs that produce
a given level of output. As it turns out, isoquants are
convex to the origin because their slope, called the marginal
rate of technical substitution (MRTS), diminishes as more
labor is used with less capital. The reason the slope diminishes
is because it is equal to the ratio of the marginal products
of the inputs, MP_{L}/MP_{K}, and as more
L is used, MPL declines (law of diminishing returns), as
less K is used, MPK increases (law of diminishing returns
in reverse!).
Next we take a look at some other properties of an isoquant
model and introduce the concept of production in the long
run.
Continues...
