Chapter Twenty-One
Cost Curves
成本曲线
Structure
? Types of cost curves
– Fixed,variable and total cost functions
– Average fixed,average variable and
average cost functions
? Marginal cost functions
? Marginal and variable cost functions
? Marginal and average cost functions
? Short run and long run cost curves
Types of Cost Curves
?A total cost curve （ 总成本曲线） is
the graph of a firm’s total cost
function.
?A variable cost curve （ 可变成本曲线
） is the graph of a firm’s variable
cost function.
?An average total cost curve （ 平均成
本曲线） is the graph of a firm’s
average total cost function.
Types of Cost Curves
?An average variable cost curve （ 平
均可变成本曲线） is the graph of a
firm’s average variable cost function.
?An average fixed cost curve （ 平均固
定成本曲线） is the graph of a firm’s
average fixed cost function.
?A marginal cost curve （ 边际成本曲线
） is the graph of a firm’s marginal
cost function.
Types of Cost Curves
?How are these cost curves related to
each other?
?How are a firm’s long-run and short-
run cost curves related?
Fixed,Variable & Total Cost Functions
?F is the total cost to a firm of its short-
run fixed inputs （ 固定投入）, 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 （ 可变投入） when
producing y output units,cv(y) is the
firm’s variable cost function.
?cv(y) depends upon the levels of the
fixed inputs.
Fixed,Variable & Total Cost Functions
?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 yv( ) ( ).? ?
y
$
F
y
$
cv(y)
y
$
F
cv(y)
y
$
F
cv(y)
c(y)
F
c y F c yv( ) ( )? ?
Av,Fixed,Av,Variable & Av,Total
Cost Curves
?The firm’s total cost function is
For y > 0,the firm’s average total
cost function is
c y F c yv( ) ( ).? ?AC y
F
y
c y
y
A F C y AVC y
v( ) ( )
( ) ( ).
? ?
? ?
Av,Fixed,Av,Variable & Av,Total
Cost Curves
?What does an average fixed cost
curve look like?
?AFC(y) is a rectangular hyperbola so
its graph looks like,..
A F C y F
y
( ) ?
$/output unit
AFC(y)
y0
AFC(y) ? 0 as y ???
Av,Fixed,Av,Variable & Av,Total
Cost Curves
?In a short-run with a fixed amount of
at least one input,the Law of
Diminishing (Marginal) Returns must
apply,causing the firm’s average
variable cost of production to
increase eventually.
$/output unit
AVC(y)
y0
$/output unit
AFC(y)
AVC(y)
y0
Av,Fixed,Av,Variable & Av,Total
Cost Curves
?And ATC(y) = AFC(y) + AVC(y)
$/output unit
AFC(y)
AVC(y)
ATC(y)
y0
ATC(y) = AFC(y) + AVC(y)
$/output unit
AFC(y)
AVC(y)
ATC(y)
y0
AFC(y) = ATC(y) - AVC(y)
AFC
$/output unit
AFC(y)
AVC(y)
ATC(y)
y0
Since AFC(y) ? 0 as y ???,
ATC(y) ? AVC(y) as y ????
AFC
$/output unit
AFC(y)
AVC(y)
ATC(y)
y0
Since AFC(y) ? 0 as y ???,
ATC(y) ? AVC(y) as y ????
And since short-run AVC(y) must
eventually increase,ATC(y) must
eventually increase in a short-run.
Marginal Cost Function
?Marginal cost is the rate-of-change of
variable production cost as the
output level changes,That is,
MC y c y
y
v( ) ( ),? ?
?
Marginal Cost Function
?The firm’s total cost function is
and the fixed cost F does not change
with the output level y,so
?MC is the slope of both the variable
cost and the total cost functions.
c y F c yv( ) ( )? ?
MC y c y
y
c y
y
v( ) ( ) ( ),? ??
?
?
?
Marginal and Variable Cost Functions
?Since MC(y) is the derivative of cv(y),
cv(y) must be the integral of MC(y),
That is,MC y
c y
y
v( ) ( )? ?
?? ? ?c y MC z dz
v
y
( ) ( ),
0
Marginal and Variable Cost Functions
MC(y)
y0
c y MC z dzv
y
( ) ( )? ? ?
?
0
?y
Area is the variable
cost of making y’ units
$/output unit
Marginal & Average Cost Functions
?How is marginal cost related to
average variable cost?
Marginal & Average Cost Functions
Since AVC y
c y
y
v( ) ( ),?
?
?
AVC y
y
y MC y c y
y
v( ) ( ) ( ),? ? ? ?1
2
Marginal & Average Cost Functions
Since AVC y
c y
y
v( ) ( ),?
?
?
AVC y
y
y MC y c y
y
v( ) ( ) ( ),? ? ? ?1
2
Therefore,?
?
AVC y
y
( ) ??
?
0y MC y c yv?
?
?
?
( ) ( ).
as
Marginal & Average Cost Functions
Since AVC y
c y
y
v( ) ( ),?
?
?
AVC y
y
y MC y c y
y
v( ) ( ) ( ),? ? ? ?1
2
Therefore,?
?
AVC y
y
( ) ??
?
0y MC y c yv?
?
?
?
( ) ( ).
as
MC y
c y
y
AVC yv( )
( )
( ).
?
?
?
?
as
?
?
AVC y
y
( ) ?
?
?
0
$/output unit
y
AVC(y)
MC(y)
$/output unit
y
AVC(y)
MC(y)
MC y AVC y AVC yy( ) ( ) ( )? ? ?? ? 0
$/output unit
y
AVC(y)
MC(y)
MC y AVC y AVC yy( ) ( ) ( )? ? ?? ? 0
$/output unit
y
AVC(y)
MC(y)
MC y AVC y AVC yy( ) ( ) ( )? ? ?? ? 0
$/output unit
y
AVC(y)
MC(y)
MC y AVC y AVC yy( ) ( ) ( )? ? ?? ? 0
The short-run MC curve intersects
the short-run AVC curve from
below at the AVC curve’s
minimum.
Marginal & Average Cost Functions
Similarly,since A TC y
c y
y
( ) ( ),??
?
A TC y
y
y MC y c y
y
( ) ( ) ( ),? ? ? ?1
2
Marginal & Average Cost Functions
Similarly,since A TC y
c y
y
( ) ( ),??
?
A TC y
y
y MC y c y
y
( ) ( ) ( ),? ? ? ?1
2
Therefore,?
?
A TC y
y
( ) ??
?
0y MC y c y?
?
?
?
( ) ( ).
as
Marginal & Average Cost Functions
Similarly,since A TC y
c y
y
( ) ( ),??
?
A TC y
y
y MC y c y
y
( ) ( ) ( ),? ? ? ?1
2
Therefore,?
?
A TC y
y
( ) ??
?
0y MC y c y?
?
?
?
( ) ( ).
as
MC y c y
y
A TC y( ) ( ) ( ).
?
?
?
?
as
?
?
A TC y
y
( ) ??
?
0
$/output unit
y
MC(y)
ATC(y)
MC y A TC y( ) ( )
?
?
?as
?
?
A TC y
y
( ) ??
?
0
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
y
AVC(y)
MC(y)
ATC(y)
Short-Run & Long-Run Total Cost
Curves
?A firm has a different short-run total
cost curve for each possible short-
run circumstance.
?Suppose the firm can be in one of
just three short-runs;
x2 = x2?
or x2 = x2?? x2? < x2?? < x2???.
or x2 = x2???.
y0
F? = w2x2?
F?
cs(y;x2?)
$
y
F?
0
F? = w2x2?
F??
F??= w2x2??
cs(y;x2?)
cs(y;x2??)
$
y
F?
0
F? = w2x2?
F??= w2x2??
A larger amount of the fixed
input increases the firm’s
fixed cost.
cs(y;x2?)
cs(y;x2??)
$
F??
y
F?
0
F? =
w2x2?F??= w2x2??
A larger amount of the fixed
input increases the firm’s
fixed cost.
Why does
a larger amount of
the fixed input reduce the
slope of the firm’s total cost curve?
cs(y;x2?)
cs(y;x2??)
$
F??
MP1 is the marginal physical productivity
of the variable input 1 (MP1=dy/dx1),so
one extra unit of input 1 gives MP1 extra
output units.
Therefore,the extra amount of input 1
needed for 1 extra output unit (dy=1) is
Short-Run & Long-Run Total Cost
Curves
units of input 1,
111/d x M P?
MP1 is the marginal physical productivity
of the variable input 1 (MP1=dy/dx1),so
one extra unit of input 1 gives MP1 extra
output units.
Therefore,the extra amount of input 1
needed for 1 extra output unit (dy=1) is
Short-Run & Long-Run Total Cost
Curves
units of input 1,
Each unit of input 1 costs w1,so the firm’s
extra cost from producing one extra unit
of output is
111/d x M P?
MC wMP? 1
1
.
Short-Run & Long-Run Total Cost
Curves
MC wMP? 1
1is the slope of the firm’s total cost curve.
Short-Run & Long-Run Total Cost
Curves
MC wMP? 1
1is the slope of the firm’s total cost curve.
If input 2 is a complement to input 1 then
MP1 is higher for higher x2.
Hence,MC is lower for higher x2.
That is,a short-run total cost curve starts
higher and has a lower slope if x2 is larger.
y
F?
0
F? =
w2x2?F??= w2x2??
F???
F???= w2x2???
cs(y;x2???)
cs(y;x2?)
cs(y;x2??)
$
F??
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?
y
F?
0
F???
y? y??
For 0 ? y ? y?,choose x2 = x2?.
cs(y;x2???)
cs(y;x2?)
cs(y;x2??)
$
F??
y
F?
0
F???
y? y??
For 0 ? y ? y?,choose x2 = x2?.
For y?? y ? y??,choose x2 = x2??.
cs(y;x2???)
cs(y;x2?)
cs(y;x2??)
$
F??
y
F?
0
F???
cs(y;x2???)
y? y??
For 0 ? y ? y?,choose x2 = x2?.
For y?? y ? y??,choose x2 = x2??.
For y??? y,choose x2 = x2???.
cs(y;x2?)
cs(y;x2??)
$
F??
y
F?
0
cs(y;x2?)
cs(y;x2??)
F???
cs(y;x2???)
y? y??
For 0 ? y ? y?,choose x2 = x2?.
For y?? y ? y??,choose x2 = x2??.
For y??? y,choose x2 = x2???.
c(y),the
firm’s long-
run total
cost curve.
$
F??
Short-Run & Long-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 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.
$
y
F?
0
F???
cs(y;x2?)
cs(y;x2??)
cs(y;x2???) c(y)
F??
Short-Run & Long-Run Average Total
Cost Curves
?For any output level y,the long-run
total cost curve always gives the lowest
possible total production cost.
?Therefore,the long-run average total
cost curve must always give the lowest
possible average 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.
Short-Run & Long-Run Average Total
Cost Curves
?E.g,suppose again that the firm can
be in one of just three short-runs;
x2 = x2?
or x2 = x2?? (x2? < x2??< x2???)
or x2 = x2???
then the firm’s three short-run
average total cost curves are,..
y
$/output unit
ACs(y;x2???)
ACs(y;x2??)
ACs(y;x2?)
Short-Run & Long-Run Average Total
Cost Curves
?The firm’s long-run average total
cost curve is the lower envelope of
the short-run average total cost
curves,..
y
$/output unit
ACs(y;x2???)
ACs(y;x2??)
ACs(y;x2?)
AC(y)The long-run av,total costcurve is the lower envelope
of the short-run av,total 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?
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.
Short-Run & Long-Run Marginal Cost
Curves
?The firm’s three short-run average
total cost curves are,..
y
$/output unit
ACs(y;x2???)
ACs(y;x2??)
ACs(y;x2?)
y
$/output unit
ACs(y;x2???)
ACs(y;x2??)
ACs(y;x2?)
MCs(y;x2?) MCs(y;x2??)
MCs(y;x2???)
y
$/output unit
ACs(y;x2???)
ACs(y;x2??)
ACs(y;x2?)
MCs(y;x2?) MCs(y;x2??)
MCs(y;x2???) AC(y)
y
$/output unit
ACs(y;x2???)
ACs(y;x2??)
ACs(y;x2?)
MCs(y;x2?) MCs(y;x2??)
MCs(y;x2???) AC(y)
y
$/output unit
ACs(y;x2???)
ACs(y;x2??)
ACs(y;x2?)
MCs(y;x2?) MCs(y;x2??)
MCs(y;x2???)
MC(y),the long-run marginal
cost curve.
Short-Run & Long-Run Marginal Cost
Curves
?For any output level y > 0,the long-
run marginal cost of production
equals to the short-run marginal cost
chosen by the firm.
A Short-Run Chosen by the Firm
? For the continuous case,where x2 can be
fixed at any value --
? At an output y*
? Chosen input,x2* = x2(y * )
and fix it to get a short run.
? At this x2*, c(y*)=cs(y*,x2*)
? Accordingly,
AC(y*)=ACs(y*,x2*)
MC(y*)=MCs(y*,x2*)
Short-Run & Long-Run Total Costs
y
$
c(y)
cs(y)
y*
Short-Run & Long-Run Marginal Cost
Curves
AC(y)
$/output unit
y
SRACs
Short-Run & Long-Run Marginal Cost
Curves
AC(y)
$/output unit
y
SRMCs
Short-Run & Long-Run Marginal Cost
Curves
AC(y)
MC(y)$/output unit
y
SRMCs
?For each y > 0,the long-run MC equals the
MC for the short-run chosen by the firm.
Cost Curves
成本曲线
Structure
? Types of cost curves
– Fixed,variable and total cost functions
– Average fixed,average variable and
average cost functions
? Marginal cost functions
? Marginal and variable cost functions
? Marginal and average cost functions
? Short run and long run cost curves
Types of Cost Curves
?A total cost curve （ 总成本曲线） is
the graph of a firm’s total cost
function.
?A variable cost curve （ 可变成本曲线
） is the graph of a firm’s variable
cost function.
?An average total cost curve （ 平均成
本曲线） is the graph of a firm’s
average total cost function.
Types of Cost Curves
?An average variable cost curve （ 平
均可变成本曲线） is the graph of a
firm’s average variable cost function.
?An average fixed cost curve （ 平均固
定成本曲线） is the graph of a firm’s
average fixed cost function.
?A marginal cost curve （ 边际成本曲线
） is the graph of a firm’s marginal
cost function.
Types of Cost Curves
?How are these cost curves related to
each other?
?How are a firm’s long-run and short-
run cost curves related?
Fixed,Variable & Total Cost Functions
?F is the total cost to a firm of its short-
run fixed inputs （ 固定投入）, 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 （ 可变投入） when
producing y output units,cv(y) is the
firm’s variable cost function.
?cv(y) depends upon the levels of the
fixed inputs.
Fixed,Variable & Total Cost Functions
?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 yv( ) ( ).? ?
y
$
F
y
$
cv(y)
y
$
F
cv(y)
y
$
F
cv(y)
c(y)
F
c y F c yv( ) ( )? ?
Av,Fixed,Av,Variable & Av,Total
Cost Curves
?The firm’s total cost function is
For y > 0,the firm’s average total
cost function is
c y F c yv( ) ( ).? ?AC y
F
y
c y
y
A F C y AVC y
v( ) ( )
( ) ( ).
? ?
? ?
Av,Fixed,Av,Variable & Av,Total
Cost Curves
?What does an average fixed cost
curve look like?
?AFC(y) is a rectangular hyperbola so
its graph looks like,..
A F C y F
y
( ) ?
$/output unit
AFC(y)
y0
AFC(y) ? 0 as y ???
Av,Fixed,Av,Variable & Av,Total
Cost Curves
?In a short-run with a fixed amount of
at least one input,the Law of
Diminishing (Marginal) Returns must
apply,causing the firm’s average
variable cost of production to
increase eventually.
$/output unit
AVC(y)
y0
$/output unit
AFC(y)
AVC(y)
y0
Av,Fixed,Av,Variable & Av,Total
Cost Curves
?And ATC(y) = AFC(y) + AVC(y)
$/output unit
AFC(y)
AVC(y)
ATC(y)
y0
ATC(y) = AFC(y) + AVC(y)
$/output unit
AFC(y)
AVC(y)
ATC(y)
y0
AFC(y) = ATC(y) - AVC(y)
AFC
$/output unit
AFC(y)
AVC(y)
ATC(y)
y0
Since AFC(y) ? 0 as y ???,
ATC(y) ? AVC(y) as y ????
AFC
$/output unit
AFC(y)
AVC(y)
ATC(y)
y0
Since AFC(y) ? 0 as y ???,
ATC(y) ? AVC(y) as y ????
And since short-run AVC(y) must
eventually increase,ATC(y) must
eventually increase in a short-run.
Marginal Cost Function
?Marginal cost is the rate-of-change of
variable production cost as the
output level changes,That is,
MC y c y
y
v( ) ( ),? ?
?
Marginal Cost Function
?The firm’s total cost function is
and the fixed cost F does not change
with the output level y,so
?MC is the slope of both the variable
cost and the total cost functions.
c y F c yv( ) ( )? ?
MC y c y
y
c y
y
v( ) ( ) ( ),? ??
?
?
?
Marginal and Variable Cost Functions
?Since MC(y) is the derivative of cv(y),
cv(y) must be the integral of MC(y),
That is,MC y
c y
y
v( ) ( )? ?
?? ? ?c y MC z dz
v
y
( ) ( ),
0
Marginal and Variable Cost Functions
MC(y)
y0
c y MC z dzv
y
( ) ( )? ? ?
?
0
?y
Area is the variable
cost of making y’ units
$/output unit
Marginal & Average Cost Functions
?How is marginal cost related to
average variable cost?
Marginal & Average Cost Functions
Since AVC y
c y
y
v( ) ( ),?
?
?
AVC y
y
y MC y c y
y
v( ) ( ) ( ),? ? ? ?1
2
Marginal & Average Cost Functions
Since AVC y
c y
y
v( ) ( ),?
?
?
AVC y
y
y MC y c y
y
v( ) ( ) ( ),? ? ? ?1
2
Therefore,?
?
AVC y
y
( ) ??
?
0y MC y c yv?
?
?
?
( ) ( ).
as
Marginal & Average Cost Functions
Since AVC y
c y
y
v( ) ( ),?
?
?
AVC y
y
y MC y c y
y
v( ) ( ) ( ),? ? ? ?1
2
Therefore,?
?
AVC y
y
( ) ??
?
0y MC y c yv?
?
?
?
( ) ( ).
as
MC y
c y
y
AVC yv( )
( )
( ).
?
?
?
?
as
?
?
AVC y
y
( ) ?
?
?
0
$/output unit
y
AVC(y)
MC(y)
$/output unit
y
AVC(y)
MC(y)
MC y AVC y AVC yy( ) ( ) ( )? ? ?? ? 0
$/output unit
y
AVC(y)
MC(y)
MC y AVC y AVC yy( ) ( ) ( )? ? ?? ? 0
$/output unit
y
AVC(y)
MC(y)
MC y AVC y AVC yy( ) ( ) ( )? ? ?? ? 0
$/output unit
y
AVC(y)
MC(y)
MC y AVC y AVC yy( ) ( ) ( )? ? ?? ? 0
The short-run MC curve intersects
the short-run AVC curve from
below at the AVC curve’s
minimum.
Marginal & Average Cost Functions
Similarly,since A TC y
c y
y
( ) ( ),??
?
A TC y
y
y MC y c y
y
( ) ( ) ( ),? ? ? ?1
2
Marginal & Average Cost Functions
Similarly,since A TC y
c y
y
( ) ( ),??
?
A TC y
y
y MC y c y
y
( ) ( ) ( ),? ? ? ?1
2
Therefore,?
?
A TC y
y
( ) ??
?
0y MC y c y?
?
?
?
( ) ( ).
as
Marginal & Average Cost Functions
Similarly,since A TC y
c y
y
( ) ( ),??
?
A TC y
y
y MC y c y
y
( ) ( ) ( ),? ? ? ?1
2
Therefore,?
?
A TC y
y
( ) ??
?
0y MC y c y?
?
?
?
( ) ( ).
as
MC y c y
y
A TC y( ) ( ) ( ).
?
?
?
?
as
?
?
A TC y
y
( ) ??
?
0
$/output unit
y
MC(y)
ATC(y)
MC y A TC y( ) ( )
?
?
?as
?
?
A TC y
y
( ) ??
?
0
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
y
AVC(y)
MC(y)
ATC(y)
Short-Run & Long-Run Total Cost
Curves
?A firm has a different short-run total
cost curve for each possible short-
run circumstance.
?Suppose the firm can be in one of
just three short-runs;
x2 = x2?
or x2 = x2?? x2? < x2?? < x2???.
or x2 = x2???.
y0
F? = w2x2?
F?
cs(y;x2?)
$
y
F?
0
F? = w2x2?
F??
F??= w2x2??
cs(y;x2?)
cs(y;x2??)
$
y
F?
0
F? = w2x2?
F??= w2x2??
A larger amount of the fixed
input increases the firm’s
fixed cost.
cs(y;x2?)
cs(y;x2??)
$
F??
y
F?
0
F? =
w2x2?F??= w2x2??
A larger amount of the fixed
input increases the firm’s
fixed cost.
Why does
a larger amount of
the fixed input reduce the
slope of the firm’s total cost curve?
cs(y;x2?)
cs(y;x2??)
$
F??
MP1 is the marginal physical productivity
of the variable input 1 (MP1=dy/dx1),so
one extra unit of input 1 gives MP1 extra
output units.
Therefore,the extra amount of input 1
needed for 1 extra output unit (dy=1) is
Short-Run & Long-Run Total Cost
Curves
units of input 1,
111/d x M P?
MP1 is the marginal physical productivity
of the variable input 1 (MP1=dy/dx1),so
one extra unit of input 1 gives MP1 extra
output units.
Therefore,the extra amount of input 1
needed for 1 extra output unit (dy=1) is
Short-Run & Long-Run Total Cost
Curves
units of input 1,
Each unit of input 1 costs w1,so the firm’s
extra cost from producing one extra unit
of output is
111/d x M P?
MC wMP? 1
1
.
Short-Run & Long-Run Total Cost
Curves
MC wMP? 1
1is the slope of the firm’s total cost curve.
Short-Run & Long-Run Total Cost
Curves
MC wMP? 1
1is the slope of the firm’s total cost curve.
If input 2 is a complement to input 1 then
MP1 is higher for higher x2.
Hence,MC is lower for higher x2.
That is,a short-run total cost curve starts
higher and has a lower slope if x2 is larger.
y
F?
0
F? =
w2x2?F??= w2x2??
F???
F???= w2x2???
cs(y;x2???)
cs(y;x2?)
cs(y;x2??)
$
F??
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?
y
F?
0
F???
y? y??
For 0 ? y ? y?,choose x2 = x2?.
cs(y;x2???)
cs(y;x2?)
cs(y;x2??)
$
F??
y
F?
0
F???
y? y??
For 0 ? y ? y?,choose x2 = x2?.
For y?? y ? y??,choose x2 = x2??.
cs(y;x2???)
cs(y;x2?)
cs(y;x2??)
$
F??
y
F?
0
F???
cs(y;x2???)
y? y??
For 0 ? y ? y?,choose x2 = x2?.
For y?? y ? y??,choose x2 = x2??.
For y??? y,choose x2 = x2???.
cs(y;x2?)
cs(y;x2??)
$
F??
y
F?
0
cs(y;x2?)
cs(y;x2??)
F???
cs(y;x2???)
y? y??
For 0 ? y ? y?,choose x2 = x2?.
For y?? y ? y??,choose x2 = x2??.
For y??? y,choose x2 = x2???.
c(y),the
firm’s long-
run total
cost curve.
$
F??
Short-Run & Long-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 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.
$
y
F?
0
F???
cs(y;x2?)
cs(y;x2??)
cs(y;x2???) c(y)
F??
Short-Run & Long-Run Average Total
Cost Curves
?For any output level y,the long-run
total cost curve always gives the lowest
possible total production cost.
?Therefore,the long-run average total
cost curve must always give the lowest
possible average 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.
Short-Run & Long-Run Average Total
Cost Curves
?E.g,suppose again that the firm can
be in one of just three short-runs;
x2 = x2?
or x2 = x2?? (x2? < x2??< x2???)
or x2 = x2???
then the firm’s three short-run
average total cost curves are,..
y
$/output unit
ACs(y;x2???)
ACs(y;x2??)
ACs(y;x2?)
Short-Run & Long-Run Average Total
Cost Curves
?The firm’s long-run average total
cost curve is the lower envelope of
the short-run average total cost
curves,..
y
$/output unit
ACs(y;x2???)
ACs(y;x2??)
ACs(y;x2?)
AC(y)The long-run av,total costcurve is the lower envelope
of the short-run av,total 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?
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.
Short-Run & Long-Run Marginal Cost
Curves
?The firm’s three short-run average
total cost curves are,..
y
$/output unit
ACs(y;x2???)
ACs(y;x2??)
ACs(y;x2?)
y
$/output unit
ACs(y;x2???)
ACs(y;x2??)
ACs(y;x2?)
MCs(y;x2?) MCs(y;x2??)
MCs(y;x2???)
y
$/output unit
ACs(y;x2???)
ACs(y;x2??)
ACs(y;x2?)
MCs(y;x2?) MCs(y;x2??)
MCs(y;x2???) AC(y)
y
$/output unit
ACs(y;x2???)
ACs(y;x2??)
ACs(y;x2?)
MCs(y;x2?) MCs(y;x2??)
MCs(y;x2???) AC(y)
y
$/output unit
ACs(y;x2???)
ACs(y;x2??)
ACs(y;x2?)
MCs(y;x2?) MCs(y;x2??)
MCs(y;x2???)
MC(y),the long-run marginal
cost curve.
Short-Run & Long-Run Marginal Cost
Curves
?For any output level y > 0,the long-
run marginal cost of production
equals to the short-run marginal cost
chosen by the firm.
A Short-Run Chosen by the Firm
? For the continuous case,where x2 can be
fixed at any value --
? At an output y*
? Chosen input,x2* = x2(y * )
and fix it to get a short run.
? At this x2*, c(y*)=cs(y*,x2*)
? Accordingly,
AC(y*)=ACs(y*,x2*)
MC(y*)=MCs(y*,x2*)
Short-Run & Long-Run Total Costs
y
$
c(y)
cs(y)
y*
Short-Run & Long-Run Marginal Cost
Curves
AC(y)
$/output unit
y
SRACs
Short-Run & Long-Run Marginal Cost
Curves
AC(y)
$/output unit
y
SRMCs
Short-Run & Long-Run Marginal Cost
Curves
AC(y)
MC(y)$/output unit
y
SRMCs
?For each y > 0,the long-run MC equals the
MC for the short-run chosen by the firm.
