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CostMin Manual
Purpose of the Program
This module is designed to let you learn about how firms can arrange their production processes
so as to minimize the costs per unit of their products. You will be given information concerning
the prices of the available resources and about the parameters of the firm’s production function.
You will also be able to choose whether to minimize the firm’s costs in the long run (with three
variable inputs) or in the medium run (with two variable inputs). The short run, in which there is
only one variable input, offers no interesting problems in cost minimization, so the short run is
not featured in this module.
The Model
This module concerns a firm that produces toy bears and whose managers want to minimize their
costs of production. It assumes that the production process of the firm allows for a given amount
of product to be produced using infinitely many combinations of three resources. The three
resources discussed in the module are labor, materials, and capital. It also assumes that each
resource in the production process faces diminishing returns when more of the given resource is
used with fixed amounts of the other resources. In other words, it assumes that if, for example,
you keep materials and capital constant and increase the amount of labor, production of bears
will increase, but each extra unit of labor will add less and less to output. The same is true if you
change one of the other inputs while keeping labor (and the third input) constant.
Isoquants
With the assumptions just described, the production process can be represented graphically by a
set of “isoquants” as in Figure 1. Each isoquant is the set of all possible combinations of labor
and materials assuming a fixed amount of capital that permit the production of a given level of
output. In figure one the combinations for 10 and 20 units of output are shown. There are other
such curves for each possible level of production. The curves slope downward because an
increase in the amount of one input (e.g., labor) would increase production. If production is to
remain constant it is necessary to reduce the amount of some other input (e.g., capital). Since the
amount of one input increases and the amount of another one decreases, that creates a negative
(downward) slope. The slopes of these isoquants get flatter and flatter as the proportion of labor
to materials increases due to the assumption of diminishing returns.
Diminishing returns is involved in this way: If you start on a point at the upper left of the figure
on the isoquant for Q = 10 and reduce materials output by, e.g., 10 pounds, it might take an hour
of labor to make up for (keeping Q at 10) losing that much in materials. However, if you keep
reducing the amount of material and adding labor, the extra labor you are adding adds less and
less to output (it takes more and more labor to bring output up a given amount). That is
diminishing returns to labor. At the same time the materials you remove were adding more
output (as you have less in materials). That is due to diminishing returns to materials. (Strictly
speaking this explanation should include a discussion of the interactions between the marginal
products of the inputs, but a full discussion of that is outside the scope of this manual.)
Figure 1 Production Isoquants
Materials
Q = 20
Q = 10
Labor
The higher the isoquant, the higher the level of production. The reason can be seen by noting
that you can get from one isoquant to a higher one by holding an input (e.g., labor) constant
while raising the amount of the other (e.g., materials). Even with diminishing returns, raising the
amount of one input while others are constant will increase production.
There are similar isoquants for combinations of capital and labor (holding materials constant)
and for capital and materials (holding labor constant). Output combinations with all three inputs
free to vary can be shown as 3-dimensional surfaces in a 3-D graph. However, since 3-D graphs
are hard to read, it is common to use two sets of isoquants for three variable problems. One set
would show combinations of labor and materials as in Figure 1. The second set would show
combinations of labor and capital for fixed amounts of materials.
The Marginal Rate of Technical Substitution
The slopes of the isoquants in Figure 1 can be described by the term Marginal Rate of Technical
Substitution (MRTS). The MRTS is obtained by using measures of the extra output that can be
obtained from an additional amount of each individual resource. The extra output of an extra unit
of labor (holding materials and capital constant) is defined as the “Marginal Product of Labor” or
MPL. The extra output from an extra unit of materials (holding labor and capital constant) is
defined as the “Marginal Product of Material” or MPM.
The marginal rate of technical substitution between labor and materials is then defined as:
MRTSLM = MPL/MPM
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Looking at the definition it should be clear how the assumption of diminishing returns affects the
MRTSLM. If the amount of labor was increased, while the amount of materials was decreased just
enough to keep production constant, MPL would fall and MPM would rise. This combination of
effects would make the slope of the isoquant (MRTSLM) fall, so the curve gets flatter and flatter.
However, since the slope of the isoquant is negative and the marginal products are positive,
strictly speaking the MRTS values are the absolute values of the slopes of the isoquants.
Isocost Lines
In order to solve the problem of cost minimization two things are needed, a description of the
production process -- the isoquants discussed above -- and a description of the costs of using
resources. This module assumes that firms can acquire additional amounts of all resources at
fixed prices (the prices neither rise nor fall as the firms buys/hires more of them). As a result, for
a fixed level of cost, the combinations of resources that the firm can get can be described (in the
two dimensional case) as a straight line, an isocost line, such as the ones in Figure 2. In the three
dimensional case combinations that incur the same cost would be a flat plane.
Figure 2 Isocost Lines
Material
TC2
TC1
Labor
In Figure 2 the line TC 1 represents one level of cost (e.g., $1000) and TC 2 represents a higher
level of cost (e.g., $2000). The slopes of TC1 and TC2 are the same, given the constant prices of
the inputs.
Cost Minimization
The difficulty of the problem of minimizing cost depends on how many inputs the firm can
control. In the very short run when the firm can control only a single input (labor in this module)
there really is no “problem” just a set of results. These results can be seen for the default
technology in the module in the table below. (The default technology is the one you get if you do
not choose to set parameters of the production function—described later in this manual—for
yourself.)
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Table 1 Short Run Costs
Output
3432.05
6337.13
8641.48
10689.54
12583.41
14370.59
16077.92
17722.38
19315.53
20865.70
22379.15
23860.68
Labor
Total Cost
20
60
100
140
180
220
260
300
340
380
420
460
737.60
1057.60
1377.60
1697.60
2017.60
2337.60
2657.60
2977.60
3297.60
3617.60
3937.60
4257.60
Average
Cost
0.21
0.17
0.16
0.16
0.16
0.16
0.17
0.17
0.17
0.17
0.18
0.18
Marginal
Cost
0.09
0.13
0.15
0.16
0.17
0.18
0.19
0.20
0.20
0.21
0.21
0.22
When more than one input can be varied the problem is to find the combination of inputs that
offers the lowest cost of producing a given output, or that offers the largest output for a given
total cost. How difficult that is depends, in this module, on whether the firm controls two inputs
or three (the more inputs the harder the problem, but the more effectively the firm can minimize
its costs).
In the two input case, for example, the solution to the problem of cost minimization can be found
by comparing labor and material costs within the firm (“inside”) and costs in outside markets
(“outside”). Suppose the firm currently can hire more labor for $10 an hour, and can get
materials for $20 a pound in the “outside,” labor and capital markets overall. In other words the
input price ratio, the measure of outside opportunity costs, is $10/$20 = 1/2. Thus the “outside”
opportunity cost of another pound of material is two hours of labor.
Suppose the current method the firm is using to produce (the combination of labor, materials,
and capital) creates an MRTS of one -- one pound of materials can substitute for one hour of
labor with production constant. If the firm hires another unit of labor for $10, it could use one
less pound of materials -- costing $20, and still get the same output. That means the firm was not
minimizing its cost. The firm could get the same production while spending $10 less, giving it a
lower cost per unit.
If the “inside” and “outside” opportunity costs are different that means the firm can change its
current practices and reduce its costs. If you recall the basic concept of comparative advantage,
in which differences in opportunity costs mean that there can be “gains from trade”, you can see
this as just another application of that concept. Where the firm’s opportunity costs differ from
those of others in the market (as reflected in the relative prices of the inputs), the firm can gain
by trading with others in the market.
The only circumstance in which the firm could not reduce its cost for a given level of production
is when the “inside” opportunity costs match the “outside” opportunity costs, in other words,
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when the MRTSLM is equal to the ratio of prices for the resources (PL/PM). If a firm, or you, can
find the combination of inputs that achieves that equality, there are no more potential gains from
trade, and the cost minimization problem has been solved.
The logic of the three input case is the same. However, if the firm controls three inputs it must
compare the inside and outside costs of inputs in more ways—labor versus materials, labor
versus capital, and capital versus materials. (One of these is redundant-- if you find the best
proportion of labor to materials and labor to capital you already have the best proportion of
capital to materials.)
Figure 3 Cost Minimization with 2 Variable Inputs
Materials
B
A
Q = 20
Q = 10
TC1
Labor
In Figure 3, Point B is a combination of inputs that allows the firm to produce 10 units of output
at a cost of TC1. However, at that combination the firms “inside” opportunity costs, represented
by the slope of the isoquant for 10 units, does not equal the “outside” opportunity costs,
represented by the slope of TC1. The firm can produce more, up to 20 units, by moving to Point
A -- where the tangency between TC1 and the isoquant for 20 units indicates that the “inside”
and “outside” opportunity costs are the same.
A firm that has three variable inputs is looking for the combination that matches “inside” and
“outside” opportunity costs in more respects. In the previous example the best point was one
where the ratio between labor and materials prices matched the ratio between the marginal
products of those two inputs. If the firm also controlled how much capital it used, it would want
this equality to hold, but it would also want the ratio between the price of labor and the price of
capital to equal the ratio of the marginal products of those two inputs (MRTSLK).
In Figure 4a, the firm is at the tangency for labor and materials, just as in Point A in Figure 3.
However it is not at the similar tangency A’ in Figure 4b, so it is not minimizing all the costs
under its control. Instead, since it is not minimizing its costs, the firm is a point like Point B,
producing less at the current cost than the amount it would get at A’, and it does not have the
lowest possible unit cost of production. The choice of Point B in the graph has the marginal
product of capital too low compared to the marginal product of labor – too much capital is being
used relative to labor. This is arbitrary; it is equally true that Point C in the graph shows a
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“wrong” combination, too little capital and too much labor. Only at A’ is there no way to reduce
the cost of producing the given output by varying the proportions of capital and labor.
Figure 4
Cost Minimization with 3 Variable Inputs
(a)
Materials
(b)
Capital
A
B
A’
C
Labor
Labor
Cost Curves
In the long run a firm’s marginal and average costs can be expressed in terms of long run cost
curves. Because this module features a firm that has constant returns to scale its lowest possible
average and marginal costs do not change as the level of production change. In other words the
firm has constant (unit) costs. This is shown by the horizontal LRAC, LRMC line in Figure 5.
A firm that controls all of its costs (in other words one that has no fixed costs because it has no
fixed inputs) could produce any amount at a cost per unit shown by that line.
In the short run the firm has short run average and marginal cost curves whose positions are
determined by the amount of the fixed input that they have. In this module choosing to control
only two inputs puts you into this short run, with capital as the fixed input that you cannot
change. For one particular value of capital the firm would have as potential least average and
marginal cost levels those indicated by the SRAC and SRMC curves. A smaller value for capital
would create similar curves, but they would be more to the left. More capital would create
similar curves to the right.
It is vital to realize that input choices that do not minimize either long or short run cost would
create costs that are not on any of these cost curves. In Figure 5 a firm would be able to produce
output Q1 at an average cost of A if it could minimize long run costs. It could produce at B if it
could only minimize short run cost. If it used a combination that did neither it would create a
cost/output point such as Point C.
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Figure 5 Short and Long Run Costs
$
B
A
C
SRMC
SRAC
●
LRAC=LRMC
Q1
Quantity
Running the Module
When you begin this module you need to make some choices that will set up the costminimizing problem for you. First, you need to decide whether you will work with two variable
inputs or three variable inputs. Once you have decided on that you have to decide whether to
have the module use the default values for the parameters or set the values of the parameters (a
through d) of the production function yourself. If you set the parameters yourself there are limits
to the values these parameters can have. The values of a, b, and c have to lie strictly between
zero and one (greater than zero and less than one) and they must add up (a + b + c) to one. The
value of d must lie strictly between 0.1 and 1.
Once you have decided on the parameters and on whether you wish to solve the two or three
variable input problem, you get to decide on how much of each (variable) input you will have the
firm hire. Once you have entered values for either two or three inputs, depending on your choice
of problems, you can get the numerical results. Note: If you chose the two input problem the
amount of capital your firm uses will be automatically set at a given value of about 4,600 hours
(expressed as 4.6 since everything is measured in thousands). If you enter 1 as your labor input
that would be 1,000 labor hours, and entering 1 as your materials input would be 1,000 pounds.
You will get the marginal products of all three inputs, one or two MRTS values, total costs, unit
costs, and marginal cost by two or three different measures (MC of increasing output if you hire
more labor, or if you buy more materials, or -- in the three input case -- if you lease more
capital). You also get reminded of the resource prices.
When you click on “graph” you will see the isocost-isoquant graph, in the three input case you
will see the capital-labor isoquants (given the amount of materials you chose). You can then
select Materials Graph on the toolbar and see the labor-materials isoquants (for the given amount
of capital). In the two input case you will only have the labor-material isoquant to look at. If you
click on “Continue” after seeing the isoquants you can see a graph showing how the optimal
(horizontal) long run AC curve compares with the AC curve generated by one of your choices --
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in effect a short run AC curve in which capital is fixed and the other inputs are adjusted in
optimal ways.
At this point you can go back and find a new, better, combination. You can determine which way
to change the input combination by comparing the price ratio for the inputs (or the price ratios in
the three input case) with the MRTS (or with the two of them in the three input case).
Additional Definitions
Total cost = cost of capital plus the cost of labor plus the cost of materials
= (PK * quantity of capital)+(PL * amount of labor)+ (PM *amount of materials)
Unit (average) cost is total cost divided by the number of units produced,
AC = Total Cost/Q
Marginal cost is the EXTRA cost paid out to cover the production of the last unit produced.
MC = ∆Total cost/∆Bears
When there are multiple variable inputs how cost changes depends on which way output is
increased (which input was changed).
Marginal Cost is determined by the largest ratio of input price to input marginal product.
If you have found the best (least cost) combination then:
MC
= PL /(∆Bears/∆Labor) = PL/MPL
= PM /(∆Bears/∆Materials) = PM /MPM
= PK /(∆Bears/∆Capital) = PL /MPK
If you have not found the least cost combination then these three measures of marginal cost will
differ. The best way of increasing output would be to increase the use of the input that offers the
lowest marginal cost. To cut costs while keeping output fixed, increase use of the input with the
lowest MC and reduce use of the one with the highest MC.
The Mathematical Model
The production function in this module is a Constant Elasticity of Substitution function:
Bears = 100 [a (Capital d) + b (Labor d) + c (Materials d)]1/d
Each input amount is raise to the "d" power, while the combination in the square brackets is
raised to the (1/d) power. The parameters a, b, and c, involve the importance of the inputs in the
production process, and in costs. The value of “d” determines the degree to which the various
inputs are substitutes for each other. The exponent for the square brackets would contain another
parameter if the module did not assume constant returns to scale, that is, it would be e/d, with e
greater than one.
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The solution to the cost minimization problem depends on the MRTS values. Looking at the two
variable input problem, you need the marginal products of labor and materials. These can be
obtained by using the “partial derivatives” of Bears with respect to each input. A partial
derivative is the change in the dependant variable (Bears) when one independent variable (e.g.,
labor) changes, while the others are kept constant (same definition as the marginal product).
MPL = ∆Bears/∆Labor = b(Labor d-1)100[a (Capital d) + b (Labor d) + c (Materials d)](1/d)-1
MPM=∆Bears/∆m\Materials=c(Materialsd-1)100[a (Capital d)+b (Labor d)+c (Materials d)](1/d)-1
MPK=∆Bears/∆capital= a(Capitald-1)100[a (Capital d)+b (Labor d)+c (Materials d)](1/d)-1
There is a MRTS between labor and materials (this is the MRTS relevant to the two variable
input problem). This is:
MRTSLM = MPL/MPM = (b/c)(Labor/Materials) d-1
The solution to the two variable input problem is found when (if PL is the price of labor and PM is
the price of materials):
MRTSLM = MPL/MPM = (b/c)(Labor/Materials) d-1 = PL / PM
If you know the values of the prices and of the three parameters, b, c, and d, then this
relationship will tell you the ratio of labor to materials that will minimize the costs of production.
The actual amounts of these resources will determine how much of the product will be produced,
but it is the ratio that determines that the cost per unit is the lowest possible.
The only difference between the solution for the two variable input problem and the three
variable input problem is that in the latter the inside opportunity cost and outside opportunity
cost of labor versus capital must also be equal. (It is equally valid to state the comparison in
terms of materials versus capital; the final results will be the same in any case.) This second
condition is:
MRTSLK = MPL/MPK = (b/a)(Labor/Capital) d-1 = PL / PK
Adding this condition makes it possible to determine the lowest cost ratio of labor to capital that
the firm can use. (If you know the best labor to materials ratio and the best labor to capital ratio,
then you already know the best materials to capital ratio; no further conditions are needed to
determine that.)
If both conditions are satisfied the firm will be producing it output at a unit cost that matches the
one found on the long run average cost curve. If only the first condition is satisfied the firm is on
a short run cost curve.
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