Tutorial 8: The Costs of Production (cont.)
In this part of Tutorial 8 we turn our focus
from the short run to the long run. We faced this same choice
in Tutorial 7 on production: how do we model production
in the long run, when all inputs are variable, in a twodimensional
graph? The answer there was isoquants  a modeling
showing all the combinations of two inputs, labor and capital,
that could produce a given level of output. [We conveniently
removed Resources from our list of inputs to make this modeling
easier  no results change when we remove it from consideration.]
So now we must develop a model of the cost of production
when the firm has two variable inputs.
Cost in the Long Run
When all inputs are variable, the firm can choose its inputs
such that costs are minimized for a given level of output.
In developing this cost model we make two simplifying assumptions:
 There are only two variable inputs, L (hours/yr) and
K (machine hours/yr); and
 Both can be hired in competitive markets, L @ $w/yr,
and K @ $r/yr, so the firm takes the prices as given.
With these assumptions we can build a model that looks
at the cost of production when the firm hires two variable
inputs.
The Isocost Line
We can construct a graphical model that shows all possible
combinations of L and K the firm can hire for a given total
cost. We start by writing out a total cost equation similar
to what we used in part a, only this time both inputs are
variable. It simply states that the total cost of production,
C, is the sum of expenditures on labor, w · L, and
expenditures on capital, r · K;
C = w·L + r·K.
This three variable equation can be written in a form
that will plot out as a straight line in a twodimensional
graph by solving for K in terms of C and L (you'll see why
in just a minute):
r · K = C  w · L
K(C, L) = (C/r)  (w/r) · L.
If we fix the amount of money the firm has to spend on
a given level of output, C_{1}, then we have a single
equation with a single unknown [remember, w and r are known
to us]:
K(L  C_{1}) = (C_{1}/r)
 (w/r) · L.
This function reveals that the amount of capital a firm
may hire at a given total cost, C_{1}, is equal
to C_{1}/r, if the firm hires no labor, and that
amount is reduced by (w/r) for every additional worker hired.
Figure 5 shows this graphically.
Figure 5
Let C_{1} = $20, w = $2 per labor
hour, and r = $2 per capital hour. Look at the vertical
intercept. If the firm only hires capital, then, at $4 per
hour, they will only be able to afford 10 machine hours
per day with the $20 per day they have to spend. Thus the
vertical intercept shows the amount of K the firm can afford
to hire if it hires no L.
For each additional labor hour hired, the
firm must give up $2/$2 = 1 machine hour of capital. (Remember
the slope of the line is w/r.) Thus the slope tells
the firm how many units of K they must trade to free up
enough money to buy 1 more unit of L.
Now look at the horizontal intercept. It tells
a similar story. If the firm only hires labor, then, at
$4 per hour, they will only be able to afford 10 labor hours
per day with the $20 per day they have to spend. Thus the
horizontal intercept shows the amount of L the firm can
afford to hire if it hires no K.
In between the intercepts lie all the combinations
of labor and capital for which the following cost equation
holds:
$20 = $2 · L + $2 · K.
Suppose the amount of money available to hire inputs rises.
That means that the amount of K the firm could hire, with
L = 0, increases, raising the vertical intercept. It also
means that the amount of L the firm could hire, with K =
0, increases, raising the horizontal intercept. The isocost
line shifts outward parallel to the original. Figure 6 shows
this for C = $30.
Figure 6
Why doesn't the slope change? The slope is
equal to w/r. Because the wage rate and the rental rate
on capital have not changed, the slope of the isocost line
does not change. That means the firm must still give up
1 unit of K in order to free up enough funds to buy one
more unit of L.
Now it's time to
"do the thing".
Click on the following link
to download the Minimizing
Production Costs Workbook. Work through
Tutorial 8 Question 1. This will help to improve
your understanding of how the isocost line
is used to model production costs when all
inputs are variable (in the long run).
Return here when you have finished.
Need help
downloading the Excel file? 

This second part of Tutorial 8 presents a discussion of
the properties of the production costs faced by firms in
the long run, when no inputs are fixed. In it we learned
that a linear equation, called an isocost line can
be used to model the total cost of hiring two variable inputs.
The next step is to combine this isocost model together
with the isoquant model developed in Tutorial 7 and use
it to see how a firm can minimize the cost of producing
a given level of output using two variable inputs.
Continues...
