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.
Long-run costs
slide 1
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.
Long-run costs
slide 2
Some isoquants: Q1 < Q2 < Q3
capital
(K)
Q3
Q2
Q1
labor (L)
Long-run costs
slide 3
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.
Long-run costs
slide 4
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?
Long-run costs
slide 5
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".
Long-run costs
slide 6
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.
Long-run costs
slide 7
capital
(K)
The slope of the isocost
line is (PL /PK )
C"/ PK
C"/ PL
Long-run costs
labor (L)
slide 8
capital
(K)
If Q* is to be produced,
what's the lowest
possible cost of
production?
C"/ PK
Q*
C"/ PL
Long-run costs
labor (L)
slide 9
capital
(K)
Exactly C* is the minimum
cost of producing Q*.
What's the rule? How
much L and K are used?
C*/ PK
C"/ PK
Q*
C"/ PL C*/ PL
Long-run costs
labor (L)
slide 10
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.
Long-run costs
slide 12
capital
(K)
Q* and C* are one point of
the firm's LRTC curve.
What's the cost of
producing Q' (>Q*)?
C*/ PK
K*
Q'
Q*
L*
Long-run costs
C*/ PL
labor (L)
slide 13
Use these points to start
plotting the LRTC curve.
capital
(K) C'/ P
K
C*/ PK
K*
Q'
Q*
L*
Long-run costs
C*/ PL
labor (L)
slide 14
Sketch in the LRTC curve.
TC
LRTC
C'
C*
Q*
Long-run costs
Q'
Q
slide 15
From this LRTC curve you can find
the corresponding average and
marginal cost curves.
Long-run costs
slide 16
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.
Long-run costs
slide 17
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.
Long-run costs
slide 18
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 shortrun cost curve to another.
Long-run costs
slide 19
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:
$/Q
Each small U-shaped
curve is a SAC curve.
LRAC
The LRAC
curve.
Long-run costs
Average costs for a
typical firm.
Q
slide 20
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.
Long-run costs
slide 21
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.
Long-run costs
slide 22
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.
Long-run costs
slide 23
ECONOMIES OF SCALE: When output
increases, long-run average costs decline.
$/Q
Long-run costs
LRAC shows
economies of
scale here.
Average costs for a
typical pizza firm.
LRAC
Q
slide 24
DISECONOMIES OF SCALE: When output
increases, long-run average costs increase.
$/Q
Long-run costs
LRAC shows
diseconomies of
scale here.
Average costs for a
typical pizza firm.
LRAC
Q
slide 25
For the U-shaped long-run average cost curve,
there are economies of scale over small
outputs, and diseconomies of scale at larger
outputs.
Long-run costs
slide 26
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.
Long-run costs
slide 27
Naturally monopolies have long-run average
cost curves that look like this:
$/Q
LRAC
Q
Electric power generation
in a local market
Long-run costs
slide 28
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.
Long-run costs
slide 29