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Intermediate Microeconomics
W3211
Lecture 15: Perfect Competition 5
The Short Run and the Long Run
Introduction
Columbia University, Spring 2016
Mark Dean: [email protected]
The Story So Far….
•
We have now modeled the perfectly competitive firm in
some detail
•
Set up the firm’s problem
•
Discussed how to split the problem into two
•
•
Today
•
Think more about the behavior of the firm in the short and
the long run
Cost minimization
Profit maximization
•
Solved both parts
•
Thought a bit about how firm behavior will change as prices
change
The Long-Run and the Short-Runs

We now introduce the distinction between long run and short
run

The long-run is the circumstance in which a firm is unrestricted in
its choice of all input levels.

The Short and the Long Run
Technology in the Short and the Long Run

In the long run a firm can choose how many workers to hire and how
many machines to use
The short-run is a circumstance in which a firm is restricted in
some way in its choice of at least one input level.

They have already purchased machines, and can now only decide
how many workers to hire

Notice that there are many possible short runs
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The Long-Run and the Short-Runs


The Long-Run and the Short-Runs
Notice, there are other possible causes of the firm being in
a ‘short run’ situation

What do short-run restrictions imply for a firm’s technology?

Suppose the short-run restriction is fixing the level of input 2.

Input 2 is thus a fixed input in the short-run. Input 1 remains
variable.
i.e. being unable to change one of its inputs:

temporarily being unable to install, or remove, machinery

being required by law to meet affirmative action quotas

having to meet domestic content regulations.
The Long-Run and the Short-Runs
The Long-Run and the Short-Runs
3 1/3
y  x1/
1 x2
y  x11/ 3101/ 3
is the long-run production
function (both x1 and x2 are variable).
y  x11/ 3 51/ 3
The short-run production function when
x2  1 is
y  x11/ 3 21/ 3
y  x11/ 311/ 3
y
y  x11 / 3 11 / 3  x11 / 3 .
The short-run production function when
x2  10 is
y  x11/3101/3.
x1
Four short-run production functions
Short-Run & Long-Run Total Costs
The Short and the Long Run

In the long-run a firm can vary all of its input levels.

Consider a firm that cannot change its input 2 level from x2’
units.

How does the short-run total cost of producing y output units
compare to the long-run total cost of producing y units of
output?
Cost Minimization in the Short and the Long Run
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Short-Run & Long-Run Total Costs

13
The long-run cost-minimization problem is
min p1 x1  p2 x2
x1 , x 2  0
subject to
f (x1, x 2 )  y.
Short-Run & Long-Run Total Costs

The short-run cost-min. problem is the long-run problem
subject to the extra constraint that x2 = x2’.

How does this affect costs?

If the long-run choice for x2 was x2’ then the extra constraint x2
= x2’ is not really a constraint at all


The short-run cost-minimization problem is
min p1 x1  p 2 x2
x1  0
subject to

f (x1, x2 )  y.
Short-Run & Long-Run Total Costs
x2
y 
Consider three output levels.
y 
Long-run and short-run total costs of producing y output units are
the same.
But, if the long-run choice for x2  x2’ then the extra constraint
x2 = x2’ prevents the firm from achieving its long-run
production cost

Short-run total cost exceed the long-run total cost of producing y
output units.
Short-Run & Long-Run Total Costs
x2
y 
y 
y
y
In the long-run when the firm
is free to choose both x1 and
x2, the least-costly input
bundles are ...
x1
x1
Short-Run & Long-Run Total Costs
x2
y 
y 
y
Short-Run & Long-Run Total Costs
x2
Long-run
output
expansion
path
y 
y
x 
2
x 2
x 2
Long-run costs are:
y 
Long-run
output
expansion
path
c( y  )  w 1x1  w 2x 2
c( y  )  w 1x1  w 2x 
2
c( y  )  w 1x1 w 2x 
2
x 
2
x 2
x 2
x1 x1 x1
x1
x1 x1 x1
x1
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Short-Run & Long-Run Total Costs


Now suppose the firm becomes subject to the short-run
constraint that x2 = x2“
Short-Run & Long-Run Total Costs
x2
Denote by cs(y) the corresponding short run cost function
y 
y 
Short-run
output
expansion
path
Long-run costs are:
c( y  )  w 1x1  w 2x 2
c( y  )  w 1x1  w 2x 
2
c( y  )  w 1x1 w 2x 
2
y
x 
2
x2
x2
x1 x1 x1
Short-Run & Long-Run Total Costs
x2
y 
y 
Short-run
output
expansion
path
Short-Run & Long-Run Total Costs
Long-run costs are:
c( y  )  w 1x1  w 2x 2
c( y  )  w 1x1  w 2x 
2
c( y  )  w 1x1 w 2x 
2
x2
y
y 
Short-run
output
expansion
path
Long-run costs are:
c( y  )  w 1x1  w 2x 2
c( y  )  w 1x1  w 2x 
2
c( y  )  w 1x1 w 2x 
2
Short-run costs are:
c s ( y  )  c( y  )
x 
2
x2
x2
x1 x1 x1
x1 x1 x1
x1
Short-Run & Long-Run Total Costs
x2
y 
y
x 
2
x2
x2
x1
y 
y 
Short-run
output
expansion
path
Short-Run & Long-Run Total Costs
Long-run costs are:
c( y  )  w 1x1  w 2x 2
c( y  )  w 1x1  w 2x 
2
c( y  )  w 1x1 w 2x 
2
x2
Short-run costs are:
y
c s ( y  )  c( y  )
c s ( y  )  c ( y  )
x 
2
x2
x2
x1 x1 x1
x1
x1
y 
y 
Short-run
output
expansion
path
Long-run costs are:
c( y  )  w 1x1  w 2x 2
c( y  )  w 1x1  w 2x 
2
c( y  )  w 1x1 w 2x 
2
Short-run costs are:
y
c s ( y  )  c( y  )
c s ( y  )  c ( y  )
x 
2
x2
x2
x1 x1 x1
x1
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Short-Run & Long-Run Total Costs
y 
x2
y 
Short-run
output
expansion
path
Long-run costs are:
c( y  )  w 1x1  w 2x 2
c( y  )  w 1x1  w 2x 
2
c( y  )  w 1x1 w 2x 
2
Short-run costs are:
y
Short-Run & Long-Run Total Costs

Short-run total cost exceeds long-run total cost except for the
output level where the short-run input level restriction is the longrun input level choice.

This says that the long-run total cost curve always has one point
in common with any particular short-run total cost curve.
c s ( y  )  c( y  )
c s ( y  )  c( y  )
c s ( y  )  c( y  )
x 
2
x2
x2
x1 x1 x1
x1
Short-Run & Long-Run Total Costs
A short-run total cost curve always has
$ one point in common with the long-run
total cost curve, and is elsewhere higher
than the long-run total cost curve.
cs(y)
The Short and the Long Run
c(y)
Cost Curves in the Short and the Long Run
F
w 2x 2
y
y 
y 
y
Types of Cost Curves

We are now going to think a little bit more about the cost
curves of a firm

In order to do so, we are going to differentiate between two
different types of cost





We now have lots of different types of costs

Total vs Fixed vs Variable

Long run vs Short run

Costs vs Average Costs vs Marginal Costs
How are these cost curves related to each other?
Typically, in the long run, all costs are variable


Fixed Costs: these do not change regardless of how much the firm
produces
Variable Costs: these do change depending on how much the firm
produces
Types of Cost Curves
If the firm produces no output it uses no input
In the short run, the firm may have some fixed costs

If they are ‘forced’ to use a certain amount of one input, then they
have to pay for that input regardless of how much they produce
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Fixed, Variable & Total Cost
Functions
Fixed, Variable & Total Cost
Functions



F is the total cost to a firm of its short-run fixed inputs. F, the
firm’s fixed cost, does not vary with the firm’s output level.
cv(y) is the total cost to a firm of its variable inputs when
producing y output units. cv(y) is the firm’s variable cost
function.


What do these various cost curves look like?

Fixed

Variable

Total
cv(y) depends upon the levels of the fixed inputs.
c(y) is the total cost of all inputs, fixed and variable, when
producing y output units. c(y) is the firm’s total cost function;
c( y )  F  c v ( y ).
$
$
cv(y)
F
y
$
y
$
c(y)
cv(y)
cv(y)
c( y )  F  c v ( y )
F
F
y
F
y
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Av. Fixed, Av. Variable & Av. Total
Cost Curves

What about average costs? (Remember
Av. Fixed, Av. Variable & Av. Total
Cost Curves
)

What does an average fixed cost curve look like?
AFC( y ) 
For y > 0, the firm’s average total cost function is
F cv ( y)

y
y
 AFC( y )  AVC( y ).
AC( y ) 

$/output unit
F
y
AFC(y) is a rectangular hyperbola so its graph looks like ...
Av. Fixed, Av. Variable & Av. Total
Cost Curves

What about average variable costs?

Well, as we have seen, this will depend on whether the firm has
increasing, decreasing, or constant returns to scale

In the short run, at least some of the inputs are fixed

We therefore typically assume that there will be diminishing
returns to scale (at least eventually)
AFC(y)  0 as y 

If we fix the number of computers, at some point we will have
decreasing returns to scale if we keep adding economists
AFC(y)
0
y
Av. Fixed, Av. Variable & Av. Total
Cost Curves

Think of a Cobb Douglas Production function

If
1 then the firm will exhibit decreasing returns to scale in
the short run

i.e. if
is fixed at
)


$/output unit
AVC(y)
This is true even if the firm exhibits increasing returns to scale in
the long run

i.e
1
0
y
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$/output unit
$/output unit
AFC(y) = ATC(y) - AVC(y)
ATC(y)
AVC(y)
AVC(y)
AFC
AFC(y)
0
$/output unit
AFC(y)
y
0
$/output unit
Two things to notice:
y
Two things to notice:
1. Since AFC(y)  0 as y ,
ATC(y)  AVC(y) as y 
ATC(y)
AFC
ATC(y)
AVC(y)
AVC(y)
AFC
AFC(y)
0
$/output unit
AFC(y)
y
y
Marginal Cost Function
Two things to notice:
1. Since AFC(y)  0 as y ,
ATC(y)  AVC(y) as y 
2. since short-run AVC(y) must
eventually increase, ATC(y) must
eventually increase in a short-run
ATC(y)
AFC
0
AVC(y)

What about marginal cost?

Well, marginal fixed cost is zero

So marginal costs are just equal to marginal variable costs
MC( y ) 
 cv ( y)
.
y
AFC(y)
0
y
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Relationship Between Marginal and
Total Cost

Marginal and Variable Cost
$/output unit
Since MC(y) is the derivative of cv(y), cv(y) must be the integral
of MC(y).

y
c v ( y  )   MC( z ) dz
0
Fundamental Theorem of Calculus
MC( y ) 
MC(y)
 cv ( y)
y
Area is the variable
cost of making y’ units
0
y
y
 cv ( y )   MC ( z )dz  cv (0)
0
y
y
 c( y )   MC ( z )dz  F
0
Marginal & Average Cost Functions
Marginal & Average Cost Functions
c ( y)
AVC( y )  v
,
y
 AVC( y ) y  MC( y )  1  c v ( y )

.
y
y2
Since

How is marginal cost related to average variable cost?
Marginal & Average Cost Functions
c ( y)
Since
AVC( y )  v
,
y
 AVC( y ) y  MC( y )  1  c v ( y )

.
y
y2
Therefore,
 AVC( y ) 
0
y

as
Av. Fixed, Av. Variable & Av. Total
Cost Curves

What does this look like in practice?

To make things more interesting, let’s think about a production
function which has both increasing and decreasing returns to
scale
 c ( y)
MC( y )  v
 AVC( y ).
 y
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Av. Fixed, Av. Variable & Av. Total
Cost Curves
A Non-Concave Production Function
Production
Function is
Convex

What does this look like in practice?

To make things more interesting, let’s think about a production
function which has both increasing and decreasing returns to
scale

Implies Marginal Cost first decreases then increases
y*
Production
Function is
Concave
*
$/output unit
$/output unit
MC( y )  AVC( y ) 
 AVC( y )
0
y
MC(y)
MC(y)
AVC(y)
AVC(y)
y
y
$/output unit
$/output unit
 AVC( y )
MC( y )  AVC( y ) 
0
y
MC( y )  AVC( y ) 
 AVC( y )
0
y
MC(y)
MC(y)
AVC(y)
AVC(y)
y
y
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$/output unit
MC( y )  AVC( y ) 
Marginal & Average Cost Functions
c( y )
,
ATC( y ) 
y
 AVC( y )
0
y
The short-run MC curve intersects
the short-run AVC curve from
below at the AVC curve’s
minimum.
Similarly, since
MC(y)
 ATC( y ) y  MC( y )  1  c( y )

.
y
y2
AVC(y)
y
Marginal & Average Cost Functions
c( y )
Similarly, since
,
ATC( y ) 
y
 ATC( y ) y  MC( y )  1  c( y )

.
y
y2
 ATC( y ) 
0
y

as
$/output unit
 ATC( y ) 
0
y

as

MC( y )  ATC( y )

MC(y)
 c( y )
MC( y ) 
 ATC( y ).
 y
ATC(y)
y
Marginal & Average Cost Functions

The short-run MC curve intersects the short-run AVC curve from
below at the AVC curve’s minimum.

And, similarly, the short-run MC curve intersects the short-run
ATC curve from below at the ATC curve’s minimum.
$/output unit
MC(y)
ATC(y)

This is also intuitive

If marginal cost is below average, the average must be going down

If marginal cost is above average, the average must be going up
AVC(y)
y
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$
Short-Run & Long-Run Total Cost
Curves

cs(y;x2)
Remember, a firm has a different short-run total cost curve for
each possible short-run circumstance.


F = w2x2
By ‘circumstance’ we mean ‘level of the fixed input’
Suppose the firm can be in one of just three short-runs;
x2 = x2
x2 < x2 < x2.
or
x2 = x2
or
x2 = x2.
F
y
0
$
$
cs(y;x2)
F = w2x2
F = w2x2
F = w2x2
F = w2x2
A larger amount of the fixed
input increases the firm’s
fixed cost.
cs(y;x2)
cs(y;x2)
cs(y;x2)
F
F
F
F
y
0
$
$
cs(y;x2)
F = w2x2
F = w2x2
A larger amount of the fixed
input increases the firm’s
fixed cost.
0
F = w2x2
F = w2x2
F = w2x2
cs(y;x2)
Why does
a larger amount of the fixed
input reduce the slope of the firm’s
total cost curve?
F
F
y
0
y
cs(y;x2)
cs(y;x2)
cs(y;x2)
F
F
F
0
y
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$
Short-Run & Long-Run Total Cost
Curves

The firm has three short-run total cost curves.

In the long-run the firm is free to choose amongst these three
since it is free to select x2 equal to any of x2, x2, or x2.

How does the firm make this choice?
For 0  y  y, choose x2 = ?
cs(y;x2)
cs(y;x2)
cs(y;x2)
F
F
F
0
$
y
y
y
$
For 0  y  y, choose x2 = x2.
For 0  y  y, choose x2 = x2.
cs(y;x2)
cs(y;x2)
For y  y  y, choose x2 = ?
cs(y;x2)
cs(y;x2)
cs(y;x2)
cs(y;x2)
F
F
F
F
F
F
0
y
y
y
0
$
y
y
$
For 0  y  y, choose x2 = x2.
For 0  y  y, choose x2 = x2.
cs(y;x2)
For y  y  y, choose x2 = x2.
cs(y;x2)
For y  y  y, choose x2 = x2.
For y  y, choose x2 = ?
cs(y;x2)
cs(y;x2)
cs(y;x2)
cs(y;x2)
F
F
F
F
F
F
0
y
y
y
y
0
y
y
y
13
4/3/2016
$
$
For 0  y  y, choose x2 = x2.
For 0  y  y, choose x2 = x2.
cs(y;x2)
For y  y  y, choose x2 = x2.
For y  y, choose x2 = x2.
For y  y, choose x2 = x2.
cs(y;x2)
F
F
F
F
F
y
y
y
Short-Run & Long-Run Total Cost
Curves

cs(y;x2)
cs(y;x2)
cs(y;x2)
F
0
cs(y;x2)
For y  y  y, choose x2 = x2.
0
c(y), the
firm’s longrun total
cost curve.
y
y
y
Short-Run & Long-Run Total Cost
Curves
If input 2 is available in continuous amounts then there is an
infinity of short-run total cost curves but the long-run total cost
curve is still the lower envelope of all of the short-run total cost
curves.
The firm’s long-run total cost curve consists of the lowest parts of
the short-run total cost curves. The long-run total cost curve is
the lower envelope of the short-run total cost curves.

$
Short-Run & Long-Run Average Total
Cost Curves
cs(y;x2)
cs(y;x2)
cs(y;x2)
c(y)
F

For any output level y, the long-run total cost curve always
gives the lowest possible total production cost.

Therefore, the long-run av. total cost curve must always give
the lowest possible av. total production cost.

The long-run av. total cost curve must be the lower envelope of
all of the firm’s short-run av. total cost curves.
F
F
0
y
14
4/3/2016
Short-Run & Long-Run Average Total
Cost Curves

$/output unit
E.g. suppose again that the firm can be in one of just three
short-runs;
x2 = x2
(x2 < x2 < x2)
or
x2 = x2
or
x2 = x2
then the firm’s three short-run average total cost curves are ...
ACs(y;x2)
ACs(y;x2)
ACs(y;x2)
y
Short-Run & Long-Run Average Total
Cost Curves

$/output unit
ACs(y;x2)
The firm’s long-run average total cost curve is the lower
envelope of the short-run average total cost curves ...
ACs(y;x2)
ACs(y;x2)
The long-run av. total cost
curve is the lower envelope
of the short-run av. total cost curves.
AC(y)
y
Short-Run & Long-Run Marginal Cost
Curves

Q: Is the long-run marginal cost curve the lower envelope of the
firm’s short-run marginal cost curves?
Short-Run & Long-Run Marginal Cost
Curves

Q: Is the long-run marginal cost curve the lower envelope of the
firm’s short-run marginal cost curves?

A: No.
15
4/3/2016
Short-Run & Long-Run Marginal Cost
Curves

$/output unit
ACs(y;x2)
ACs(y;x2)
The firm’s three short-run average total cost curves are ...
ACs(y;x2)
y
$/output unit
MCs(y;x2)
MCs(y;x2)
$/output unit
ACs(y;x2)
MCs(y;x2)
ACs(y;x2)
ACs(y;x2)
ACs(y;x2)
ACs(y;x2)
ACs(y;x2)
MCs(y;x2)
MCs(y;x2)
y
+
MCs(y;x2)
Short-Run & Long-Run Marginal Cost
Curves

Below y’, will choose
′, despite the fact it has higher marginal
cost than either ′′ or ′′′

Because it gives lower total cost
y
y
y
$/output unit
AC(y)
MCs(y;x2)
MCs(y;x2)
ACs(y;x2)
ACs(y;x2)
ACs(y;x2)
MCs(y;x2)
y
y
AC(y)
y
16
4/3/2016
$/output unit
MCs(y;x2)
MCs(y;x2)
ACs(y;x2)
ACs(y;x2)
Short-Run & Long-Run Marginal Cost
Curves

ACs(y;x2)
For any output level y > 0, the long-run marginal cost of
production is the marginal cost of production for the short-run
chosen by the firm.
MCs(y;x2)
MC(y), the long-run marginal
cost curve.
y
$/output unit
MCs(y;x2)
MCs(y;x2)
Short-Run & Long-Run Marginal Cost
ACs(y;x2)
ACs(y;x2)

For any output level y > 0, the long-run marginal cost is the
marginal cost for the short-run chosen by the firm.

This is always true, no matter how many and which short-run
circumstances exist for the firm.
ACs(y;x2)
MCs(y;x2)
MC(y), the long-run marginal
cost curve.
y
Short-Run & Long-Run Marginal Cost
Short-Run & Long-Run Marginal Cost
$/output unit

For any output level y > 0, the long-run marginal cost is the
marginal cost for the short-run chosen by the firm.

So for the continuous case, where x2 can be fixed at any value
of zero or more, the relationship between the long-run marginal
cost and all of the short-run marginal costs is ...
SRACs
AC(y)
y
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4/3/2016
Short-Run & Long-Run Marginal Cost
$/output unit
Short-Run & Long-Run Marginal Cost
$/output unit
SRMCs
SRMCs
AC(y)
MC(y)
AC(y)
y
y
For each y > 0, the long-run MC equals the
MC for the short-run chosen by the firm.
Summary
•
Today we focused on the difference between the short
and the long run
•
•
How technology changes
How costs change
Summary
18