Chapter Twenty
Cost Minimization
成本最小化
Structure
?The cost minimization problem
?Average costs
?Returns to scale and total and
average costs
?Short run and long run costs
Cost Minimization
?A firm is a cost-minimizer if it
produces any given output level y ? 0
at smallest possible total cost.
?c(y) denotes the firm’s smallest
possible total cost for producing y
units of output.
?c(y) is the firm’s total cost function
（ 总成本函数）,
Cost Minimization
?When the firm faces given input
prices w = (w1,w2,…,w n) the total cost
function will be written as
c(w1,…,w n,y).
The Cost-Minimization Problem
?Consider a firm using two inputs to
make one output.
?The production function is
y = f(x1,x2).
?Take the output level y ? 0 as given.
?Given the input prices w1 and w2,the
cost of an input bundle (x1,x2) is
w1x1 + w2x2.
The Cost-Minimization Problem
?For given w1,w2 and y,the firm’s
cost-minimization problem is to
solve m in,x x w x w x
1 2 0
1 1 2 2? ?
subject to f x x y(,),1 2 ?
The Cost-Minimization Problem
?The levels x1*(w1,w2,y) and x1*(w1,w2,y)
in the least-costly input bundle are the
firm’s conditional demands for inputs
1 and 2 （ 条件要素需求）,
?The (smallest possible) total cost for
producing y output units is therefore
c w w y w x w w y
w x w w y
(,,) (,,)
(,,).
*
*
1 2 1 1 1 2
2 2 1 2
?
?
Conditional Input Demands
?Given w1,w2 and y,how is the least
costly input bundle located?
?And how is the total cost function （
成本函数） computed?
Iso-cost Lines （ 等成本线）
?A curve that contains all of the input
bundles that cost the same amount
is an iso-cost curve.
?E.g.,given w1 and w2,the $100 iso-
cost line has the equation
w x w x1 1 2 2 100? ?,
Iso-cost Lines
?Generally,given w1 and w2,the
equation of the $c iso-cost line is
i.e.
?Slope is - w1/w2.
x w
w
x c
w2
1
2
1
2
? ? ?,
w x w x c1 1 2 2? ?
Iso-cost Lines
c’ ? w1x1+w2x2
c” ? w1x1+w2x2
c’ < c”
x1
x2 Slopes = -w1/w2.
The y’-Output Unit Isoquant
x1
x2 All input bundles yielding y’ units
of output,Which is the cheapest?
f(x1,x2) ? y’
The Cost-Minimization Problem
x1
x2 All input bundles yielding y’ units
of output,Which is the cheapest?
f(x1,x2) ? y’
The Cost-Minimization Problem
x1
x2 All input bundles yielding y’ units
of output,Which is the cheapest?
f(x1,x2) ? y’
x1*
x2*
The Cost-Minimization Problem
x1
x2
f(x1,x2) ? y’
x1*
x2*
At an interior cost-min input bundle:
(a) f x x y(,)* *1 2 ? ?
The Cost-Minimization Problem
x1
x2
f(x1,x2) ? y’
x1*
x2*
At an interior cost-min input bundle:
(a) and
(b) slope of isocost = slope of
isoquant; i.e.
f x x y(,)* *1 2 ? ?
? ? ? ?ww TR S MPMP at x x1
2
1
2
1 2(,).* *
A Cobb-Douglas Example of Cost
Minimization
?A firm’s Cobb-Douglas production
function is
?Input prices are w1 and w2.
?What are the firm’s conditional input
demand functions?
y f x x x x? ?(,),/ /1 2 11 3 22 3
A Cobb-Douglas Example of Cost
Minimization
At the input bundle (x1*,x2*) which minimizes
the cost of producing y output units:
(a)
(b)
y x x? ( ) ( )* / * /1 1 3 2 2 3and
? ? ? ? ?
? ?
?
?
w
w
y x
y x
x x
x x
x
x
1
2
1
2
1
2 3
2
2 3
1
1 3
2
1 3
2
1
1 3
2 3
2
? ?
? ?
/
/
( / )( ) ( )
( / )( ) ( )
.
* / * /
* / * /
*
*
A Cobb-Douglas Example of Cost
Minimization
y x x? ( ) ( )* / * /1 1 3 2 2 3
w
w
x
x
1
2
2
12
?
*
*,(a) (b)
From (b),x
w
w x2
1
2
1
2* *,?
Now substitute into (a) to gety x w
w
x w
w
x? ?
??
?
??
? ?
??
?
??
( ),* / *
/ /
*
1
1 3 1
2
1
2 3
1
2
2 3
1
2 2
x w
w
y1 2
1
2 3
2
*
/
? ?
??
?
??So is the firm’s conditional
demand for input 1.
A Cobb-Douglas Example of Cost
Minimization
x ww x2 1
2
1
2* *? x w
w
y1 2
1
2 3
2
*
/
? ?
??
?
??
is the firm’s conditional demand for input 2.
Since and
x w
w
w
w
y w
w
y2 1
2
2
1
2 3
1
2
1 32
2
2* / /? ?
??
?
??
? ?
??
?
??
A Cobb-Douglas Example of Cost
Minimization
So the cheapest input bundle yielding y
output units is
? ?x w w y x w w y
w
w
y
w
w
y
1 1 2 2 1 2
2
1
2 3
1
2
1 3
2
2
* *
/ /
(,,),(,,)
,.?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
x
2
x
1
Fixed w1 and w2.
Conditional Input Demand Curves
?y??y
???y
x
2
x
1
Fixed w1 and w2.
Conditional Input Demand Curves
x y1*( )?
x y2*( )? ?y??y???y
?y
?y
x y2*( )?
x y1*( )?
x2*
x1*
y
y
x
2
x
1
Fixed w1 and w2.
Conditional Input Demand Curves
x y1*( )?x y
1*( )??
x y2* ( )??x y
2*( )? ?y??y???y
??y
??y
?y
?y
x y2*( )??x y2
*( )?
x y1*( )??x y1
*( )?
x2*
x1*
y
y
x
2
x
1
Fixed w1 and w2.
Conditional Input Demand Curves
x y1*( )???
x y2* ( )???
x y1*( )?x y
1*( )??
x y2* ( )??x y
2*( )? ?y??y???y
???y
???y
??y
??y
?y
?y
x y2*( )???x y
2*( )??
x y2*( )?
x y1*( )???x y
1*( )??
x y1*( )?
x2*
x1*
y
y
x
2
x
1
Fixed w1 and w2.
Conditional Input Demand Curves
x y1*( )???
x y2* ( )???
x y1*( )?x y
1*( )??
x y2* ( )??x y
2*( )?
output
expansion
path
?y??y
???y
???y
???y
??y
??y
?y
?y
x y2*( )???x y
2*( )??
x y2*( )?
x y1*( )???x y
1*( )??
x y1*( )?
Cond,demand
for
input 2
Cond.
demand
for
input 1
x2*
x1*
y
y
A Cobb-Douglas Example of Cost
Minimization
For the production function
the cheapest input bundle yielding y output
units is
? ?x w w y x w w y
w
w
y
w
w
y
1 1 2 2 1 2
2
1
2 3
1
2
1 3
2
2
* *
/ /
(,,),(,,)
,.?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
3/223/1121 xx)x,x(fy ??
A Cobb-Douglas Example of Cost
Minimization
c w w y w x w w y w x w w y
w
w
w
y w
w
w
y
w w y w w y
w w
y
(,,) (,,) (,,)
.
* *
/ /
/
/ / / / /
/
1 2 1 1 1 2 2 2 1 2
1
2
1
2 3
2
1
2
1 3
2 3
1
1 3
2
2 3 1 3
1
1 3
2
2 3
1 2
2
1 3
2
2
1
2
2
3
4
? ?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
?
?
?
So the firm’s total cost function is
A Perfect Complements Example of
Cost Minimization
?The firm’s production function is
?Input prices w1 and w2 are given.
?What are the firm’s conditional
demands for inputs 1 and 2?
?What is the firm’s total cost
function?
y x x? m in {,}.4 1 2
A Perfect Complements Example of
Cost Minimization
x1
x2
x1*
= y/4
x2* = y
4x1 = x2
min{4x1,x2} ? y’
Where is the least costly
input bundle yielding
y’ output units?
A Perfect Complements Example of
Cost Minimization
y x x? m in {,}4 1 2The firm’s production function is
and the conditional input demands arex w w y y
1 1 2 4* (,,) ?x w w y y2 1 2* (,,),?and
So the firm’s total cost function is
c w w y w x w w y
w x w w y
w
y
w y
w
w y
(,,) (,,)
(,,)
.
*
*
1 2 1 1 1 2
2 2 1 2
1 2
1
2
4 4
?
?
? ? ? ?
?
?
?
?
?
?
Average Total Production Costs
?For positive output levels y,a firm’s
average total cost of producing y
units isAC w w y c w w y
y
(,,) (,,),1 2 1 2?
Returns-to-Scale and Av,Total Costs
?The returns-to-scale properties of a
firm’s technology determine how
average production costs change with
output level.
?Our firm is presently producing y’
output units.
?How does the firm’s average
production cost change if it instead
produces 2y’ units of output?
Constant Returns-to-Scale and Average
Total Costs
?If a firm’s technology exhibits
constant returns-to-scale then
doubling its output level from y’ to
2y’ requires doubling all input levels,
?Total production cost doubles.
?Average production cost does not
change.
Decreasing Returns-to-Scale and
Average Total Costs
?If a firm’s technology exhibits
decreasing returns-to-scale then
doubling its output level from y’ to
2y’ requires more than doubling all
input levels,
?Total production cost more than
doubles.
?Average production cost increases.
Increasing Returns-to-Scale and
Average Total Costs
?If a firm’s technology exhibits
increasing returns-to-scale then
doubling its output level from y’ to
2y’ requires less than doubling all
input levels,
?Total production cost less than
doubles.
?Average production cost decreases.
Returns-to-Scale and Av,Total Costs
y
$/output unit
constant r.t.s.
decreasing r.t.s.
increasing r.t.s.
AC(y)
Returns-to-Scale and Total Costs
?What does this imply for the shapes
of total cost functions?
Returns-to-Scale and Total Costs
y
$ c(y)
y’ 2y’
c(y’)
c(2y’) Slope = c(2y’)/2y’
= AC(2y’).
Slope = c(y’)/y’
= AC(y’).
Av,cost increases with y if the firm’s
technology exhibits decreasing r.t.s.
Returns-to-Scale and Total Costs
y
$ c(y)
y’ 2y’
c(y’)
c(2y’)
Slope = c(2y’)/2y’
= AC(2y’).
Slope = c(y’)/y’
= AC(y’).
Av,cost decreases with y if the firm’s
technology exhibits increasing r.t.s.
Returns-to-Scale and Total Costs
y
$ c(y)
y’ 2y’
c(y’)
c(2y’)
=2c(y’) Slope = c(2y’)/2y’
= 2c(y’)/2y’
= c(y’)/y’
so
AC(y’) = AC(2y’).
Av,cost is constant when the firm’s
technology exhibits constant r.t.s.
Short-Run & Long-Run Total Costs
?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?
Short-Run & Long-Run Total Costs
?The long-run cost-minimization
problem is
?The short-run cost-minimization
problem is
m in
,x x
w x w x
1 2 0
1 1 2 2? ?
subject to f x x y(,),1 2 ?
m in
x
w x w x
1 0
1 1 2 2? ? ?
subject to f x x y(,),1 2? ?
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’.
?If the long-run choice for x2 was x2’
then the extra constraint x2 = x2’ is not
really a constraint at all and so the
long-run and short-run total costs of
producing y output units are the same.
Short-Run & Long-Run Total Costs
?But,if the long-run choice for x2 ? x2”
then the extra constraint x2 = x2”
prevents the firm in this short-run from
achieving its long-run production cost,
causing the short-run total cost to
exceed the long-run total cost of
producing y output units.
Short-Run & Long-Run Total Costs
x1
x2
???y
??y
?y
Consider three output levels.
Short-Run & Long-Run Total Costs
x1
x2
???y
??y
?y
In the long-run when the firm
is free to choose both x1 and
x2,the least-costly input
bundles are,..
Short-Run & Long-Run Total Costs
x1
x2
???y
??y
?y
?x1 ??x1 ???x1
?x2??x2
???x2
Long-run
output
expansion
path
Short-Run & Long-Run Total Costs
x1
x2
???y
??y
?y
Long-run
output
expansion
path
?x1 ??x1 ???x1
?x2??x2
???x2
Long-run costs are:
c y w x w x
c y w x w x
c y w x w x
( )
( )
( )
? ? ? ? ?
?? ? ?? ? ???
??? ? ???? ???
1 1 2 2
1 1 2 2
1 1 2 2
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
x1
x2
???y
??y
?y
?x1 ??x1 ???x1
?x2??x2
???x2
Short-run
output
expansion
path
Long-run costs are:
c y w x w x
c y w x w x
c y w x w x
( )
( )
( )
? ? ? ? ?
?? ? ?? ? ???
??? ? ???? ???
1 1 2 2
1 1 2 2
1 1 2 2
Short-Run & Long-Run Total Costs
x1
x2
???y
??y
?y
?x1 ??x1 ???x1
?x2??x2
???x2
Short-run
output
expansion
path
Long-run costs are:
c y w x w x
c y w x w x
c y w x w x
( )
( )
( )
? ? ? ? ?
?? ? ?? ? ???
??? ? ???? ???
1 1 2 2
1 1 2 2
1 1 2 2
Short-run costs are:c y c ys ( ) ( )? ? ?
Short-Run & Long-Run Total Costs
x1
x2
???y
??y
?y
?x1 ??x1 ???x1
?x2??x2
???x2
Short-run
output
expansion
path
Long-run costs are:
c y w x w x
c y w x w x
c y w x w x
( )
( )
( )
? ? ? ? ?
?? ? ?? ? ???
??? ? ???? ???
1 1 2 2
1 1 2 2
1 1 2 2
Short-run costs are:c y c y
c y c y
s
s
( ) ( )
( ) ( )
? ? ?
?? ? ??
Short-Run & Long-Run Total Costs
x1
x2
???y
??y
?y
?x1 ??x1 ???x1
?x2??x2
???x2
Short-run
output
expansion
path
Long-run costs are:
c y w x w x
c y w x w x
c y w x w x
( )
( )
( )
? ? ? ? ?
?? ? ?? ? ???
??? ? ???? ???
1 1 2 2
1 1 2 2
1 1 2 2
Short-run costs are:
c y c y
c y c y
c y c y
s
s
s
( ) ( )
( ) ( )
( ) ( )
? ? ?
?? ? ??
??? ? ???
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 long-run 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.
Short-Run & Long-Run Total Costs
y
$
c(y)
???y??y?y
cs(y)
Fw x? ??
2 2
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.
Cost Minimization
成本最小化
Structure
?The cost minimization problem
?Average costs
?Returns to scale and total and
average costs
?Short run and long run costs
Cost Minimization
?A firm is a cost-minimizer if it
produces any given output level y ? 0
at smallest possible total cost.
?c(y) denotes the firm’s smallest
possible total cost for producing y
units of output.
?c(y) is the firm’s total cost function
（ 总成本函数）,
Cost Minimization
?When the firm faces given input
prices w = (w1,w2,…,w n) the total cost
function will be written as
c(w1,…,w n,y).
The Cost-Minimization Problem
?Consider a firm using two inputs to
make one output.
?The production function is
y = f(x1,x2).
?Take the output level y ? 0 as given.
?Given the input prices w1 and w2,the
cost of an input bundle (x1,x2) is
w1x1 + w2x2.
The Cost-Minimization Problem
?For given w1,w2 and y,the firm’s
cost-minimization problem is to
solve m in,x x w x w x
1 2 0
1 1 2 2? ?
subject to f x x y(,),1 2 ?
The Cost-Minimization Problem
?The levels x1*(w1,w2,y) and x1*(w1,w2,y)
in the least-costly input bundle are the
firm’s conditional demands for inputs
1 and 2 （ 条件要素需求）,
?The (smallest possible) total cost for
producing y output units is therefore
c w w y w x w w y
w x w w y
(,,) (,,)
(,,).
*
*
1 2 1 1 1 2
2 2 1 2
?
?
Conditional Input Demands
?Given w1,w2 and y,how is the least
costly input bundle located?
?And how is the total cost function （
成本函数） computed?
Iso-cost Lines （ 等成本线）
?A curve that contains all of the input
bundles that cost the same amount
is an iso-cost curve.
?E.g.,given w1 and w2,the $100 iso-
cost line has the equation
w x w x1 1 2 2 100? ?,
Iso-cost Lines
?Generally,given w1 and w2,the
equation of the $c iso-cost line is
i.e.
?Slope is - w1/w2.
x w
w
x c
w2
1
2
1
2
? ? ?,
w x w x c1 1 2 2? ?
Iso-cost Lines
c’ ? w1x1+w2x2
c” ? w1x1+w2x2
c’ < c”
x1
x2 Slopes = -w1/w2.
The y’-Output Unit Isoquant
x1
x2 All input bundles yielding y’ units
of output,Which is the cheapest?
f(x1,x2) ? y’
The Cost-Minimization Problem
x1
x2 All input bundles yielding y’ units
of output,Which is the cheapest?
f(x1,x2) ? y’
The Cost-Minimization Problem
x1
x2 All input bundles yielding y’ units
of output,Which is the cheapest?
f(x1,x2) ? y’
x1*
x2*
The Cost-Minimization Problem
x1
x2
f(x1,x2) ? y’
x1*
x2*
At an interior cost-min input bundle:
(a) f x x y(,)* *1 2 ? ?
The Cost-Minimization Problem
x1
x2
f(x1,x2) ? y’
x1*
x2*
At an interior cost-min input bundle:
(a) and
(b) slope of isocost = slope of
isoquant; i.e.
f x x y(,)* *1 2 ? ?
? ? ? ?ww TR S MPMP at x x1
2
1
2
1 2(,).* *
A Cobb-Douglas Example of Cost
Minimization
?A firm’s Cobb-Douglas production
function is
?Input prices are w1 and w2.
?What are the firm’s conditional input
demand functions?
y f x x x x? ?(,),/ /1 2 11 3 22 3
A Cobb-Douglas Example of Cost
Minimization
At the input bundle (x1*,x2*) which minimizes
the cost of producing y output units:
(a)
(b)
y x x? ( ) ( )* / * /1 1 3 2 2 3and
? ? ? ? ?
? ?
?
?
w
w
y x
y x
x x
x x
x
x
1
2
1
2
1
2 3
2
2 3
1
1 3
2
1 3
2
1
1 3
2 3
2
? ?
? ?
/
/
( / )( ) ( )
( / )( ) ( )
.
* / * /
* / * /
*
*
A Cobb-Douglas Example of Cost
Minimization
y x x? ( ) ( )* / * /1 1 3 2 2 3
w
w
x
x
1
2
2
12
?
*
*,(a) (b)
From (b),x
w
w x2
1
2
1
2* *,?
Now substitute into (a) to gety x w
w
x w
w
x? ?
??
?
??
? ?
??
?
??
( ),* / *
/ /
*
1
1 3 1
2
1
2 3
1
2
2 3
1
2 2
x w
w
y1 2
1
2 3
2
*
/
? ?
??
?
??So is the firm’s conditional
demand for input 1.
A Cobb-Douglas Example of Cost
Minimization
x ww x2 1
2
1
2* *? x w
w
y1 2
1
2 3
2
*
/
? ?
??
?
??
is the firm’s conditional demand for input 2.
Since and
x w
w
w
w
y w
w
y2 1
2
2
1
2 3
1
2
1 32
2
2* / /? ?
??
?
??
? ?
??
?
??
A Cobb-Douglas Example of Cost
Minimization
So the cheapest input bundle yielding y
output units is
? ?x w w y x w w y
w
w
y
w
w
y
1 1 2 2 1 2
2
1
2 3
1
2
1 3
2
2
* *
/ /
(,,),(,,)
,.?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
x
2
x
1
Fixed w1 and w2.
Conditional Input Demand Curves
?y??y
???y
x
2
x
1
Fixed w1 and w2.
Conditional Input Demand Curves
x y1*( )?
x y2*( )? ?y??y???y
?y
?y
x y2*( )?
x y1*( )?
x2*
x1*
y
y
x
2
x
1
Fixed w1 and w2.
Conditional Input Demand Curves
x y1*( )?x y
1*( )??
x y2* ( )??x y
2*( )? ?y??y???y
??y
??y
?y
?y
x y2*( )??x y2
*( )?
x y1*( )??x y1
*( )?
x2*
x1*
y
y
x
2
x
1
Fixed w1 and w2.
Conditional Input Demand Curves
x y1*( )???
x y2* ( )???
x y1*( )?x y
1*( )??
x y2* ( )??x y
2*( )? ?y??y???y
???y
???y
??y
??y
?y
?y
x y2*( )???x y
2*( )??
x y2*( )?
x y1*( )???x y
1*( )??
x y1*( )?
x2*
x1*
y
y
x
2
x
1
Fixed w1 and w2.
Conditional Input Demand Curves
x y1*( )???
x y2* ( )???
x y1*( )?x y
1*( )??
x y2* ( )??x y
2*( )?
output
expansion
path
?y??y
???y
???y
???y
??y
??y
?y
?y
x y2*( )???x y
2*( )??
x y2*( )?
x y1*( )???x y
1*( )??
x y1*( )?
Cond,demand
for
input 2
Cond.
demand
for
input 1
x2*
x1*
y
y
A Cobb-Douglas Example of Cost
Minimization
For the production function
the cheapest input bundle yielding y output
units is
? ?x w w y x w w y
w
w
y
w
w
y
1 1 2 2 1 2
2
1
2 3
1
2
1 3
2
2
* *
/ /
(,,),(,,)
,.?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
3/223/1121 xx)x,x(fy ??
A Cobb-Douglas Example of Cost
Minimization
c w w y w x w w y w x w w y
w
w
w
y w
w
w
y
w w y w w y
w w
y
(,,) (,,) (,,)
.
* *
/ /
/
/ / / / /
/
1 2 1 1 1 2 2 2 1 2
1
2
1
2 3
2
1
2
1 3
2 3
1
1 3
2
2 3 1 3
1
1 3
2
2 3
1 2
2
1 3
2
2
1
2
2
3
4
? ?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
?
?
?
So the firm’s total cost function is
A Perfect Complements Example of
Cost Minimization
?The firm’s production function is
?Input prices w1 and w2 are given.
?What are the firm’s conditional
demands for inputs 1 and 2?
?What is the firm’s total cost
function?
y x x? m in {,}.4 1 2
A Perfect Complements Example of
Cost Minimization
x1
x2
x1*
= y/4
x2* = y
4x1 = x2
min{4x1,x2} ? y’
Where is the least costly
input bundle yielding
y’ output units?
A Perfect Complements Example of
Cost Minimization
y x x? m in {,}4 1 2The firm’s production function is
and the conditional input demands arex w w y y
1 1 2 4* (,,) ?x w w y y2 1 2* (,,),?and
So the firm’s total cost function is
c w w y w x w w y
w x w w y
w
y
w y
w
w y
(,,) (,,)
(,,)
.
*
*
1 2 1 1 1 2
2 2 1 2
1 2
1
2
4 4
?
?
? ? ? ?
?
?
?
?
?
?
Average Total Production Costs
?For positive output levels y,a firm’s
average total cost of producing y
units isAC w w y c w w y
y
(,,) (,,),1 2 1 2?
Returns-to-Scale and Av,Total Costs
?The returns-to-scale properties of a
firm’s technology determine how
average production costs change with
output level.
?Our firm is presently producing y’
output units.
?How does the firm’s average
production cost change if it instead
produces 2y’ units of output?
Constant Returns-to-Scale and Average
Total Costs
?If a firm’s technology exhibits
constant returns-to-scale then
doubling its output level from y’ to
2y’ requires doubling all input levels,
?Total production cost doubles.
?Average production cost does not
change.
Decreasing Returns-to-Scale and
Average Total Costs
?If a firm’s technology exhibits
decreasing returns-to-scale then
doubling its output level from y’ to
2y’ requires more than doubling all
input levels,
?Total production cost more than
doubles.
?Average production cost increases.
Increasing Returns-to-Scale and
Average Total Costs
?If a firm’s technology exhibits
increasing returns-to-scale then
doubling its output level from y’ to
2y’ requires less than doubling all
input levels,
?Total production cost less than
doubles.
?Average production cost decreases.
Returns-to-Scale and Av,Total Costs
y
$/output unit
constant r.t.s.
decreasing r.t.s.
increasing r.t.s.
AC(y)
Returns-to-Scale and Total Costs
?What does this imply for the shapes
of total cost functions?
Returns-to-Scale and Total Costs
y
$ c(y)
y’ 2y’
c(y’)
c(2y’) Slope = c(2y’)/2y’
= AC(2y’).
Slope = c(y’)/y’
= AC(y’).
Av,cost increases with y if the firm’s
technology exhibits decreasing r.t.s.
Returns-to-Scale and Total Costs
y
$ c(y)
y’ 2y’
c(y’)
c(2y’)
Slope = c(2y’)/2y’
= AC(2y’).
Slope = c(y’)/y’
= AC(y’).
Av,cost decreases with y if the firm’s
technology exhibits increasing r.t.s.
Returns-to-Scale and Total Costs
y
$ c(y)
y’ 2y’
c(y’)
c(2y’)
=2c(y’) Slope = c(2y’)/2y’
= 2c(y’)/2y’
= c(y’)/y’
so
AC(y’) = AC(2y’).
Av,cost is constant when the firm’s
technology exhibits constant r.t.s.
Short-Run & Long-Run Total Costs
?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?
Short-Run & Long-Run Total Costs
?The long-run cost-minimization
problem is
?The short-run cost-minimization
problem is
m in
,x x
w x w x
1 2 0
1 1 2 2? ?
subject to f x x y(,),1 2 ?
m in
x
w x w x
1 0
1 1 2 2? ? ?
subject to f x x y(,),1 2? ?
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’.
?If the long-run choice for x2 was x2’
then the extra constraint x2 = x2’ is not
really a constraint at all and so the
long-run and short-run total costs of
producing y output units are the same.
Short-Run & Long-Run Total Costs
?But,if the long-run choice for x2 ? x2”
then the extra constraint x2 = x2”
prevents the firm in this short-run from
achieving its long-run production cost,
causing the short-run total cost to
exceed the long-run total cost of
producing y output units.
Short-Run & Long-Run Total Costs
x1
x2
???y
??y
?y
Consider three output levels.
Short-Run & Long-Run Total Costs
x1
x2
???y
??y
?y
In the long-run when the firm
is free to choose both x1 and
x2,the least-costly input
bundles are,..
Short-Run & Long-Run Total Costs
x1
x2
???y
??y
?y
?x1 ??x1 ???x1
?x2??x2
???x2
Long-run
output
expansion
path
Short-Run & Long-Run Total Costs
x1
x2
???y
??y
?y
Long-run
output
expansion
path
?x1 ??x1 ???x1
?x2??x2
???x2
Long-run costs are:
c y w x w x
c y w x w x
c y w x w x
( )
( )
( )
? ? ? ? ?
?? ? ?? ? ???
??? ? ???? ???
1 1 2 2
1 1 2 2
1 1 2 2
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
x1
x2
???y
??y
?y
?x1 ??x1 ???x1
?x2??x2
???x2
Short-run
output
expansion
path
Long-run costs are:
c y w x w x
c y w x w x
c y w x w x
( )
( )
( )
? ? ? ? ?
?? ? ?? ? ???
??? ? ???? ???
1 1 2 2
1 1 2 2
1 1 2 2
Short-Run & Long-Run Total Costs
x1
x2
???y
??y
?y
?x1 ??x1 ???x1
?x2??x2
???x2
Short-run
output
expansion
path
Long-run costs are:
c y w x w x
c y w x w x
c y w x w x
( )
( )
( )
? ? ? ? ?
?? ? ?? ? ???
??? ? ???? ???
1 1 2 2
1 1 2 2
1 1 2 2
Short-run costs are:c y c ys ( ) ( )? ? ?
Short-Run & Long-Run Total Costs
x1
x2
???y
??y
?y
?x1 ??x1 ???x1
?x2??x2
???x2
Short-run
output
expansion
path
Long-run costs are:
c y w x w x
c y w x w x
c y w x w x
( )
( )
( )
? ? ? ? ?
?? ? ?? ? ???
??? ? ???? ???
1 1 2 2
1 1 2 2
1 1 2 2
Short-run costs are:c y c y
c y c y
s
s
( ) ( )
( ) ( )
? ? ?
?? ? ??
Short-Run & Long-Run Total Costs
x1
x2
???y
??y
?y
?x1 ??x1 ???x1
?x2??x2
???x2
Short-run
output
expansion
path
Long-run costs are:
c y w x w x
c y w x w x
c y w x w x
( )
( )
( )
? ? ? ? ?
?? ? ?? ? ???
??? ? ???? ???
1 1 2 2
1 1 2 2
1 1 2 2
Short-run costs are:
c y c y
c y c y
c y c y
s
s
s
( ) ( )
( ) ( )
( ) ( )
? ? ?
?? ? ??
??? ? ???
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 long-run 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.
Short-Run & Long-Run Total Costs
y
$
c(y)
???y??y?y
cs(y)
Fw x? ??
2 2
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.
