LONG-RUN COSTS

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LONG-RUN COSTS

• In the long-run there are no fixed inputs, and
  therefore no fixed costs. All costs are
  variable.
• Another way to look at the long-run is that in
  the long-run a firm can choose any amount of
  fixed costs it wants for making short-run
  decisions.

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Cost Minimization

• Suppose a firm has a production function with two
  variable inputs, labor (L), and capital (K).
• Q f(L, K)
• This production function can be represented by an
  isoquant map.

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Some isoquants Q1 lt Q2 lt Q3
capital (K)
labor (L)
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Marginal Rate of Technical Substitution

• The Marginal Rate of Technical Substitution of L
  or K is the amount of K it takes to make up for
  the loss of one unit of L, output constant.
• MRTS (L for K) -?K??L, output constant.
• MRTS is (minus) the slope of an isoquant.

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Problem

• The firm is told it has to produce a particular
  output level, say, Q.
• The firm can buy L and K at fixed known prices,
• PL and PK.
• How should the firm choose L and K if it must
  produce Q at minimum cost?

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• The firm's total cost is its expenditure on all
  of its inputs
• C PL L PKK
• For a given value of C, the above expression is
  known as an "isocost" line, or "cost constraint".
  

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• The firm's cost contraint can be rewritten as
• K (C/ PK ) - (PL /PK ) L
• and drawn on the same set of axes as the firm's
  isoquants.

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capital (K)
The slope of the isocost line is (PL /PK )
C"/ PK
labor (L)
C"/ PL
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If Q is to be produced, what's the lowest
possible cost of production?
capital (K)
C"/ PK
Q
labor (L)
C"/ PL
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Exactly C is the minimum cost of producing
Q. What's the rule? How much L and K are used?
capital (K)
C/ PK
C"/ PK
Q
C/ PL
labor (L)
C"/ PL
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MRTS (L for K) (PL /PK ) L and K are the best
amounts of the inputs.
capital (K)
C/ PK
K
Q
C/ PL
L
labor (L)
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Finding the Long-run Total Cost Curve

• The Long-Run Total Cost Curve shows the minimum
  cost of producing any output when all inputs are
  variable.
• In this case the number of inputs is two.
• We show how to find a couple of points on the
  firm's LRTC curve.

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Q and C are one point of the firm's LRTC
curve. What's the cost of producing Q' (gtQ)?
capital (K)
C/ PK
K
Q'
Q
C/ PL
L
labor (L)
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Use these points to start plotting the LRTC curve.
capital (K)
C'/ PK
C/ PK
K
Q'
Q
C/ PL
L
labor (L)
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Sketch in the LRTC curve.
TC
C'
C
Q
Q'
Q
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From this LRTC curve you can find the
corresponding average and marginal cost curves.
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The Long-run Average Cost Curve
The long-run average cost curve shows the minimum
average cost at each output level when all inputs
are variable, that is, when the firm can have any
plant size it wants. There is a relationship
between the LRAC curve and the firm's set of
short-run average cost curves.
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SR and LR Average Costs

• Economists use the term plant size to talk
  about having a particular amount of fixed inputs.
  Choosing a different amount of plant and
  equipment (plant size) amounts to choosing an
  amount of fixed costs.
• Economists want you to think of fixed costs as
  being associated with plant and equipment.
  Bigger plants have larger fixed costs.

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• If each plant size is associated with a different
  amount of fixed costs, then each plant size for a
  firm will give us a different set of short-run
  cost curves.
• Choosing a different plant size (a long-run
  decision) then means moving from one short-run
  cost curve to another.

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• Economists usually assume that plant size is
  infinitely divisible (variable). In the case of
  finely divisible plant size, the LRAC curve might
  look like this

Each small U-shaped curve is a SAC curve.
/Q
LRAC
The LRAC curve.
Q
Average costs for a typical firm.
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• In the preceding graph, each short-run cost curve
  corresponds to a particular amount of fixed
  inputs.
• As the fixed input amount increases in the long
  run, you move to different SR cost curves, each
  one corresponding to a particular plant size.

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• Notice in the graphs of LRAC curves presented so
  far that the curves have been drawn to be
  U-shaped. That is, when output is increasing
  LRAC at first falls, and then eventually rises.
• The overall shape of the long-run average cost
  curve depends on the technology of production.

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• For example, advantages implicit in large scale
  production (with large plants) may allow firms to
  produce large outputs at lower cost per unit.
• On the other hand, firms may get so big that ever
  increasing managerial and monitoring costs may
  cause unit costs to rise.

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ECONOMIES OF SCALE When output increases,
long-run average costs decline.
LRAC shows economies of scale here.
/Q
LRAC
Q
Average costs for a typical pizza firm.
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DISECONOMIES OF SCALE When output increases,
long-run average costs increase.
LRAC shows diseconomies of scale here.
/Q
LRAC
Q
Average costs for a typical pizza firm.
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• For the U-shaped long-run average cost curve,
  there are economies of scale over small outputs,
  and diseconomies of scale at larger outputs.

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• Not all firms necessarily suffer from
  diseconomies of scale at large outputs.
• When a firm has economies of scale over a range
  of outputs big enough to supply the total market
  demand, that firm is called a natural monopoly.

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• Naturally monopolies have long-run average cost
  curves that look like this

/Q
LRAC
Q
Electric power generation in a local market
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• As we will see, firms in perfect competition must
  have U-shaped long-run average cost curves.
• One conclusion from this is that only certain
  industries can be expected to be perfectly
  competitive. And a crucial factor is the
  technology of production, since that is what
  determines the shape of the long-run average cost
  curve.