# cost

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```					Cost
Overview
In this section we want to translate the production data
into cost data. In other words, we will want to
understand how the cost of producing various units of
output might change as different amounts of inputs are
used.
Fixed/variable inputs
Inputs can be classified as either fixed or variable.

A variable input is one that can be changed as the level
of output is changed .
A fixed input is one that can not be changed as the level
of output is changed.

We often think of labor as a variable input and capital or
land as a fixed input.
Short run/long run
The notion of a fixed or variable input is related to the
time frame of production.
The short run is that period of time when at least one
input is fixed in amount.
The long run is that period of time in which all inputs
are variable.
As an example of this consider fast food in Wayne.
About any store in town could remodel and increase
floor space in about 3 months. So after 3 months we
have the long run, all inputs can vary - even floor space.
But less than three months is the short run because there
is only so much floor space to use.
short run costs
In the short run we will consider the fixed and variable
costs of production and how they change as more of the
variable input is used.
Definitions:
Total cost (TC) = Total variable cost(TVC) + Total fixed
cost (TFC).
Marginal cost(MC) = (change in TC)/(change in output).
where change in output = 1 when possible.
Average cost (AC) = TC/Q.
Average variable cost(AVC) = TVC/Q.
Average fixed cost(AFC) = TFC/Q.
Note that in the short run fixed costs must be paid
whether output is zero, 100,000, or any other number of
units.
example
Let’s take the production example. Say the fixed costs is
\$1000 per unit of capital and we have two units and the
cost of labor is \$400 per unit.
The next screen shows the example with an added TP or
output amount.
QL    Q     MPL   APL   TFC    TVC    TC

0     0                 2000    0     2000
1     76    76    76    2000   400    2400
2    248    172   124   2000   800    2800
3    492    244   164   2000   1200   3200
4    784    292   196   2000   1600   3600
5    1100   316   220   2000   2000   4000
6    1416   316   236   2000   2400   4400
7    1708   292   244   2000   2800   4800
8    1952   244   244   2000   3200   5200
9    2124   172   236   2000   3600   5600
10   2200   76    220   2000   4000   6000
11   2156   -44   196   2000   4400   6400
QL    Q     MPL   APL   TFC    TVC    TC     AFC     AVC    ATC     MC

0     0                 2000    0     2000

1     76    76    76    2000   400    2400   26.32   5.26   31.58   5.26

2    248    172   124   2000   800    2800    8.06   3.23   11.29   2.33

3    492    244   164   2000   1200   3200    4.07   2.44     6.5   1.64

4    784    292   196   2000   1600   3600    2.55   2.04    4.59   1.37

5    1100   316   220   2000   2000   4000    1.82   1.82    3.64   1.27

6    1416   316   236   2000   2400   4400    1.41   1.69    3.11   1.27

7    1708   292   244   2000   2800   4800    1.17   1.64    2.81   1.37

8    1952   244   244   2000   3200   5200    1.02   1.64    2.66   1.64

9    2124   172   236   2000   3600   5600    0.94   1.69    2.64   2.33

10   2200   76    220   2000   4000   6000    0.91   1.82    2.73   5.26
COST Curves
Total cost curve

7000                                                                          Costs per unit
6000
Dollars of Cost

5000                                  TFC
4000
3000
TVC                      35
2000                                  TC                       30

Dollars per unit
1000
0                                                           25
0   1000       2000     3000                            20
TP or Q or output                                   15
10
5
0
0   500     1000     1500      2000   2500
Quantity of Output
Idealized graph of per unit costs
\$/unit
in the short run
AC

AVC

MC

Q
Note AVC and AC equal MC when AVC and AC are at
their minimum values.
When you look back at the marginal product and average product
curves note that the horizontal axis is measuring labor units used
and the curves are inverted u-shaped curves.
When you look back at the marginal cost and various average cost
curves note that the horizontal axis is measuring output units and
the curves are u-shaped curves.
There is a relationship between these two graphs. When you move
to the right by adding labor in the one graph you are moving to the
right in the other by having output increased.
How much output should the firm make? How much of the
variable labor should be hired?
We are not yet ready to answer this question. We are only
talking about cost. Will have to talk revenue and profit later
before we can answer these questions.
Two production facilities, one type
of output.
Say you can make output in 1 of 2 facilities. Let’s also say the
MC in each (plant a MC) MCa = 12Qa and (plant b MC) MCb
= 4Qb.
For the firm the first unit of output has MC = 12 in plant a and
4 in plant b. So the first unit should come from plant b.
For the firm the second unit could be the first from plant a or
the second from plant be. The respective MC’s are 12 and 8.
Have it come from plant b.
The third unit will also come from plant b. But the 4th unit for
the firm will be the 1st unit from plant a because of comparing
MC’s, 12 in a and 16 in b.
Two production facilities, one type
of output.
We would keep apply this logic to see which plant the next until of
output should come from.
A summary of the firm level MC is found by adding the two MC’s in
the following way:
MCa = 12Qa and MCb = 4Qb. Re-arrange these in Q from like the
following (just express each with Q term on the left) Qa = (1/12)MCa
and Qb = (1/4)MCb.
Q = Qa + Qb = (1/12)MC + (1/4)MC = ((1/12) + (3/12))MC =
(1/3)MC. So we have Q = (1/3)MC or MC = 3Q. If you want Q = 32,
MC = 96 at the firm level. IN plant a if MC = 96, Qa = (96/12) = 8 and
in plant b Qb = (96/4) = 24. Note Qa + Qb = 32.
Isocost lines
An isocost line includes all possible combinations of
labor and capital that can be purchased for a given
total cost.
In equation form the total cost is
TC = wL + rK,
where TC = Total cost,
w = the wage rate,
L = the amount of labor taken,
r = the rental price of capital, and
K = the amount of capital taken. This equation
can be re-expressed as
K = TC/r - (w/r) L.
example
As an example say labor is \$6 per unit and capital is
\$10 per unit. Then if we look at a total cost of \$100
we see various combinations of inputs:
L = 10 and K = 4 or
L = 0 and K = 10 or
L = 16.67 and K = 0, amoung others.

On the next screen we can view the isocost line in a
graph.
K   graph of isocost line
This is the isocost line
at \$100. If we wanted
to see higher costs we
would shift the line out
in a parallel shift and a
lower cost we have a
shift in.
L
K                        cost and output
On this slide I want to
concentrate on one
level of output, as
summarized by the
K1                                  isoquant. Input
combination L1, K1
K2                                  could be used and have
cost summarized by
K*                                  4th highest isocost
L shown. L2, K2 would
L1 L2    L*                 be cheaper, and L*, K*
is the lowest cost to produce the given level of output. Here
the cheapest cost of the output occurs at a tangency point.
K                       cost and output
On this slide I want to
concentrate on one
level of cost, as
summarized by the
isocost line. Input
combination L1, K1
K1                                  could be used and have
this cost but more
K*                                  output would be
L obtained if L*, K* were
L1         L*                 used.
Here, the most output for a given cost occurs at a
tangency point.
cost and output
On the last two screens we have seen the tangency of
an isoquant and isocost line shows either
1) the cheapest way to produce a certain level of
output, or
2) the most output that can be obtained for a given
amount of cost.
These two things are different sides of the same coin
and profit maximizing firms would be expected to
reach the tangency positions.
The exception to reaching the tangency would be the
short run when the amount of some input can not be
changed to reach the tangency. In the long run all
inputs can be changed in amount and thus the
tangency point could be reached.
K                                   short run
Here the cheapest way
to produce the output
level as depicted in the
isoquant would be to
K1                                hire L*, K*. But the
firm as committed to
K2                                having K1 units of
capital. Thus the cost
K*                                of this output is
L indicated by the fourth
L1 L2     L*                 highest isocost line.
We could follow K1 out and see costs of other levels of
output(by putting in more isoquants).
Tangency = equal slopes
Our optimal point for the firm in general was slope of isoquant =
slope of isocost (or MRTS = w/r) and this gives
MPl/MPk = w/r. We can rearrange this to get
MPl/w = MPk/r. This last form has an interesting economic
interpretation: The extra output from the last dollar spent on labor
has to equal the extra output from the last dollar spent on capital.
Remember this occurs at the lowest cost level.
Example: say MPl = MPk = 4, and w = 4 and r = 2. Note we do
not have MPl/w = MPk/r. Since MPl/w = 4/4 = 1/1 taking a dollar
away from labor means we lose roughly 1 unit of output. Also
since MPk/r = 4/2 = 2/1 = 1/.5 we only have to spend 50 cents on
capital to add back the unit of output we lost from having less
labor. When the ratio is not equal costs are too high!
Cost concepts in a graph
unit costs
\$            ATC1     MC          ATC

b
AVC

AVC1
a           MC1
Q

Q1
Interpretation
• I have picked Q1 arbitrarily and have drawn
a line from this Q up to the highest cost
curve.
• MC1 is the MC of the this unit.
• AVC1 is the AVC of all the units.
• ATC1 is the ATC of all the units.
Interpretation continued
• Since TC = TFC + TVC, ATC = AFC +
AVC or AFC = ATC - AVC.
• So in the diagram, AFC1 = ATC1 - AVC1.
• Area a = AVC1 times Q1 = TVC1.
• Area b = (ATC1 - AVC1) times Q1 = TFC1.
Interpretation continued
• Area a + b =TVC1 + TFC1 = TC1.
• The concept of diminishing returns is the
primary force driving costs in the short run.
The rest of the ideas are definitions. The u -
shape of the curves are due to the
diminishing returns concept.
Long Run
• The above example assumed we could only
have one unit of capital. Now let’s imagine
we can have two units of capital.
• We would have a similar table of numbers
and graphs as we did when only one unit of
capital was available.
Long Run continued
• When we switch from one unit of capital to
two units, we have the long run because all
inputs are then variable.
• But with the two units we would have short
run curves for that level of capital.
• Now we have two sets of cost curves, one
for one unit of capital and one for two units
of capital.
Long run continued
• Thus the graph of the long run is really just
a bunch of curves, one for each plant size.
• I will draw two ATC curves, each with a
different amount of capital used.
ATC   Long Run Graphs
ATC2

ATC1

Q
Q1
Interpretation
• If output is going to be less than Q1 in the
long run then only one unit of capital would
be wanted because those units would be
produced cheapest with one unit of capital.
• Greater than Q1 would be produced
cheapest with two units of capital.
Interpretation
• The long run curve is parts of the short run
curves. For each range of output the long
run curve is the segment of the short run
curve that is the lowest, representing the
cheapest way to produce that range of
output in the long run. The final long run
curve is smooth. Let’s see.
ATC
Smooth long run curve

Q
Reason for long run shape
• The long run cost curve is said to be u -
shaped, just as in the short run, but for a
different reason. In the short run we had
diminishing returns. In the long run we
have economies of scale.
Reason continued
• The basic idea of economies of scale is that
at least for a while when the plant size is
increased the average cost curve is pushed
down, implying average costs are lowest in
a bigger plant. It may be that further
increases in plant size push the average cost
curve back up. This would technically be
called diseconomies of scale.

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 views: 10 posted: 10/27/2011 language: English pages: 35