Tutorial 8: The Costs of Production (cont.)
In Tutorial
7 we saw how we could model the production process with
two variable inputs by using a graphical model of isoquants.
In the second part of Tutorial
8 we learned how we could model the cost of production
using two variable inputs with a graphical model of isocost
lines. Now it's time to put the two together to model how
a firm can choose that combination of the two variable inputs
that minimizes the cost of producing a given level of output.
Choosing Inputs
Suppose the firm's goal is to produce Q_{1} units/year.
They could hire any combination of labor and capital on
the isoquant Q_{1}. But logically the firm would
rather choose that combination that entails the minimum
cost. Stated simply, the firm's objective is to minimize
the cost of producing a given level of output.
See Figure 7. Suppose the firm's output must be Q = 100
units/day. It could choose any one of the numerous combinations
of L and K along the isoquant Q1 that would produce the
desired output. Suppose it chose (2.5, 10)? If w = $2/unit
of L and r = $2/unit of K, then that combination would cost
the firm 2.5 · $2 + 10 · $2 = $25 to produce
100 units/day. Is that the cheapest it can get? Well it
could hire no L and no K and that would cost them $0, but
the firm is constrained by its desire to produce 100 units/day.
So how about these combinations:
Table 1
Labor hours 
Capital hours 
Cost 
Comments 
3.6 
7.0 
$21.20 
Cheaper, but... 
5.0 
5.0 
$20.00 
Looking better... 
7.0 
3.6 
$21.20 
oops! 
Figure 7
Well the combination (5, 5) was cheapest. So what are the
conditions of the model at this point? Well the slopes
of the isocost and the isquant are equal there. But
why would that be the cheapest?
The absolute value of the slope of isoquant = MRTS.
The marginal rate of technical substitution reveals how
much K the firm is willing to give up when one more L
is used to produce a given level of Q. The absolute value
of the slope of isocost = w/r, which reveals how many
units of K the firm must trade to free up enough money to
buy one more unit of L. At (2.5, 10.0), MRTS > w/r
meaning firm is willing to give up more K when it hires
one more L than is required by the market. So it will happily
hire more L and reduce the amount of K used to produce Q
because it will be cheaper. This also holds true for (3.6,
7.0)  MRTS is still > w/r but not by as much. Finally,
at (5, 5) MRTS = w/r and the firm has reached the point
where further additions of L (even with reductions in K)
will raise the cost of producing 100 units of output to
more than $20.
So, mathemagically, the cost minimizing firm will vary
L* and K* until MRTS  r/w = 0 (which is the same as
saying MRTS = w/r. Why write it this way? See the Excel
Workbook for this Tutorial...). But MRTS = w/r at (0, 0).
True, but the firm is constrained by its desire to produce
a given level of output, i.e., its objective is subject
to the constraint that A*L^{a}*K^{b} = Q_{1},
where MRTS = (a/L*) / (b/K*). So, graphically, the combination
of inputs that satisfies both the objective and the constraint,
is found at the point of tangency between the isocost line
and isquant.
Caution: This model is easily confused with the
consumer choice model. The elements appear similar but their
roles differ. While there are many similarities, there is
an important difference. In the consumer choice model the
budget line represents the individual's constraint on choice.
Indifference curves "appear" through any market
basket of goods. In the production costs model, the isoquant
represents the firm's constraint on production costs. Isocost
lines "appear" through any combination of labor
and capital the firm hires.
Now it's time to
"do the thing".
Click on the following link
to download the Minimizing
Production Costs Workbook. Work through
Tutorial 8 Questions 2  7. This will help
to improve your understanding of how we combine
the isoquant and the isocost lines to model
the costminimizing choice of two variable
inputs.
Return here when you have finished.
Need help
downloading the Excel file? 

This third part of Tutorial 8 presents a discussion
of how we can combine the 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.
In part d we will use the isocost/isoquant
model to discover the differences in the cost of increasing
output in the short run (when at least one input is fixed)
and in the long run (when no inputs are fixed).
Continues...
