Production and Costs in the Long Run

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Production and Costs in the Long Run
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The long run

• The long run is the time frame longer or just as
  long as it takes to alter the plant.
• Thus the long run is that time period in which
  all inputs are variable.

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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 PLL PKK, where TC
Total cost, PL the wage rate, L the amount
of labor taken, PK the rental price of
capital, and K the amount of capital taken.
This equation can be re-expressed as K TC/ PK
- (PL/ PK) L.
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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.
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graph of isocost line
K
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
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cost and output
K
On this slide I want to concentrate on one level
of output, as summarized by the isoquant. Input
combination L1, K1 could be used and have cost
summarized by 4th highest isocost shown. L2, K2
would be cheaper, and L, K
K1
K2 K
L
L1 L2 L
is the lowest cost combination of inputs to
produce the given level of output. Here the
cheapest cost of the output occurs at a
tangency point between an isocost and isoquant.
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cost and output
K
On this slide I want to concentrate on one level
of cost, as summarized by the isocost line.
Input combination L1, K1 could be used and have
this cost but more output would be obtained if
L, K were used.
K1 K
L
L1 L
Here, the most output for a given cost occurs at
a tangency point.
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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 in the long run.
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Tangency
In the long run when a firm is able to change all
inputs we see the firm will go to a point where
the slope of an isocost line is tangent to a
isoquant. This means the slopes are equal. Thus,
slope of isocost (PL/ PK) MRTS slope of
isoquant. Remember we said MRTS can be shown to
be the ratio of marginal products of labor to
capital. Thus PL/PK MPL/MPK means MPK/PK
MPL/PL. This is a statement that the bang for
the buck should be the same for both inputs. In
other words the additional output for each input
per dollar spent should be equal across inputs
when all is said and done.
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If you happen to be at a point where the ratios
are not equal take more of the input that has the
higher ratio because its marginal product will
diminish with a greater amount taken. You will
probably have to take less of the other input.
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Expansion path
K
Once we have a unit cost of capital and labor we
can draw many isocosts, each one that is farther
out has a higher cost. We can see the tangency
of each isocost with an isoquant (output level).
L
In the long run the firm will be at one of the
points of tangency. When connect all those
points we have the expansion path. In the long
run the firm will be on the expansion path.
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The Short Run and the Long Run and seeing the
connection between the two. The exception to
reaching the tangency in the long run 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.
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short run
K
Here the cheapest way to produce the output level
as depicted in the isoquant would be to hire L,
K. But the firm has committed to having K1
units of capital. Thus the cost of this output
is indicated by the fourth highest isocost line.
K1
K2 K
L
L1 L2 L
We could follow K1 out and see costs of other
levels of output(by putting in more isoquants).
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As you follow along K1 maybe one output level
will occur where that short run point is exactly
the same as the long run point. In that one case
the cost level is the same in both the short run
and the long run. Remember the output level shown
on the previous screen in the short run with K1
has a higher cost to produce that output than
would occur in the long run. So, in the long run,
you make the cost of a certain level of output
the lowest by not only adjusting labor to the
right amount but capital to the right amount.
But, if in the short run you are not at the right
amount of capital then you will produce the
output at a higher cost because in the short run
you are stuck at a certain level of capital.
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On the next slide I have two short run average
cost curves. Each one represents the average
cost with different amounts of the fixed input
capital. SO, maybe ATC1 could have 1 unit of
capital and ATC2 could have two units of capital.
There really should be lots more of these curves
but I show two to get to the next point. If
output will be less than Q in the long run, then
in the short run costs might be too high if we
have two units of capital. But, in the long run
capital would be switched to 1 unit. Similarly,
output above Q has lowest cost when made with two
units of capital.
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To get Long Run Graphs
ATC
ATC2
ATC1
Q
Q
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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.

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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.

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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. Lets see.

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Smooth long run curve
ATC
Q
Each point on the long run curve is really just a
point off the lowest short run curve at a level
of output.
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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.

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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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Long Run

• Another way to view the long run is to think
  about different short run situations and put them
  together. Think of a short run with one capital
  unit. Think of one with two capital units, and
  so on.
• We would have a similar table of numbers and
  graphs as we did in the short run example when
  only one unit of capital was available.