Chapter Eighteen
Technology
技术
Structure
? Describing technologies
?Production set or technology set
?Production function
?Isoquant
? Marginal product
? Returns to scale
? Technical rate of substitution
? Well-behaved technologies
? Long run and short run
Technologies
?A technology is a process by which
inputs are converted to an output.
?E.g,labor,a computer,a projector,
electricity,and software are being
combined to produce this lecture.
Technologies
?Usually several technologies will
produce the same product -- a
blackboard and chalk can be used
instead of a computer and a
projector.
?Which technology is,best”?
?How do we compare technologies?
Input Bundles
?xi denotes the amount used of input i;
i.e,the level of input i.
?An input bundle is a vector of the
input levels; (x1,x2,…,x n).
Production Functions (生产函数）
?y denotes the output level.
?The technology?s production
function states the maximum amount
of output possible from an input
bundle.
y f x x n? (,,)1 ?
Production Functions
y = f(x) is the
production
function.
x? x
Input Level
Output Level
y?
y? = f(x?) is the maximal
output level obtainable
from x? input units.
One input,one output
Technology Sets
? A production plan is an input bundle and an
output level; (x1,…,x n,y).
? A production plan is feasible （ 可行） if
? The collection of all feasible production
plans is the production set (生产集 ) or
technology set （ 技术集）,
y f x x n? (,,)1 ?
Technology Sets
y = f(x) is the
production
function.
x? x
Input Level
Output Level
y?
y”
y? = f(x?) is the maximal
output level obtainable
from x? input units.
One input,one output
y” = f(x?) is an output level
that is feasible from x?
input units.
Technology Sets
The production set or technology set is T x x y y f x x a n d
x x
n n
n
? ?
? ?
{(,,,) | (,,)
,,}.
1 1
1 0 0
? ?
?
Technology Sets
x? x
Input Level
Output Level
y?
One input,one output
y”
The technology
set
Technology Sets
x? x
Input Level
Output Level
y?
One input,one output
y”
The technology
setTechnically
inefficient
plans
Technically
efficient plans
Technologies with Multiple Inputs
?What does a technology look like
when there is more than one input?
?The two input case,Input levels are
x1 and x2,Output level is y.
?Suppose the production function is
y f x x x x? ?(,),1 2 11/ 3 21/ 32
Technologies with Multiple Inputs
?E.g,the maximal output level
possible from the input bundle
(x1,x2) = (1,8) is
?And the maximal output level
possible from (x1,x2) = (8,8) is
y x x? ? ? ? ? ? ? ?2 2 1 8 2 1 2 411/ 3 21/ 3 1/ 3 1/ 3,
y x x? ? ? ? ? ? ? ?2 2 8 8 2 2 2 811/ 3 21/ 3 1/ 3 1/ 3,
Technologies with Multiple Inputs
?An isoquant (等产量线 ) is the set of
all possible combinations of inputs 1
and 2 that are just sufficient to
produce a given amount of output.
Isoquants with Two Variable Inputs
y ???
y ???
x1
x2
Technologies with Multiple Inputs
?The complete collection of isoquants
is the isoquant map.
?The isoquant map is equivalent to
the production function -- each is the
other.
?E.g,3/123/1121 2),( xxxxfy ??
Isoquants with Two Variable Inputs
y ???
y ???
x1
x2
y ???
y ???
Cobb-Douglas Technologies
?A Cobb-Douglas production function
is of the form
?E.g.
with
y A x x xa a na n? ? ?1 21 2 ?,
y x x? 11/ 3 21/ 3
n A a a n d a? ? ? ?2 1 13 131 2,,.
x2
x1
All isoquants are hyperbolic
(双曲线 ),asymptoting (渐进 ) to,
but never touching any axis.
Cobb-Douglas Technologies
y x xa a? 1 21 2
x2
x1
All isoquants are hyperbolic,
asymptoting to,but never
touching any axis.
Cobb-Douglas Technologies
x x ya a1 21 2 ? '
x x ya a1 21 2 ? "
y" y'> y x x
a a? 1 21 2
Fixed-Proportions Technologies
x2
x1
min{x1,2x2} = 14
4 8 14
24
7
min{x1,2x2} = 8
min{x1,2x2} = 4
x1 = 2x2
y x x? m i n {,}1 22
Perfect-Substitution Technologies
9
3
18
6
24
8
x1
x2
x1 + 3x2 = 18
x1 + 3x2 = 36
x1 + 3x2 = 48
All are linear and parallel
y x x? ?1 23
Marginal (Physical) Products
?The marginal product (边际产量 ) of
input i is the rate-of-change of the
output level as the level of input i
changes,holding all other input
levels fixed.
?That is,
y f x x n? (,,)1 ?
i
i x
y
MP
?
?
?
Marginal (Physical) Products
E.g,if
y f x x x x? ?(,) /1 2 11/ 3 22 3
then the marginal product of input 1 is
MP yx x x1
1
1
2 3
2
2 31
3? ?
??
?
/ /
and the marginal product of input 2 isMP y
x x x2 2 1
1/ 3
2
1/ 32
3? ?
??
?,
Marginal (Physical) Products
Typically the marginal product of one
input depends upon the amount used of
other inputs,E.g,if
MP x x1 1 2 3 22 313? ? / /then,
MP x x1 1 2 3 2 3 1 2 313 8 43? ?? ?/ / /
and if x2 = 27 then
if x2 = 8,
MP x x1 1 2 3 2 3 1 2 313 27 3? ?? ?/ / /,
Marginal (Physical) Products
?The marginal product of input i is
diminishing if it becomes smaller as
the level of input i increases,That is,
if
.02
2
????
?
?
???
?
?
iiii
i
x
y
x
y
xx
MP
?
?
?
?
?
?
?
?
Marginal (Physical) Products
MP x x1 1 2 3 22 313? ? / /MP x2 11/ 3 2 1/ 323? ?and
so ?
?
MP
x x x
1
1
1
5 3
2
2 32
9 0? ? ?
? / /
?
?
MP
x x x
2
2
11/ 3 2 4 3
2
9 0? ? ?
? /,and
Both marginal products are diminishing.
E.g,if y x x? 11/ 3 22 3/then
Returns-to-Scale（ 规模收益）
?Marginal products describe the
change in output level as a single
input level changes.
?Returns-to-scale describes how the
output level changes as all input
levels change in direct proportion
(e.g,all input levels doubled,or
halved).
Returns-to-Scale
If,for any input bundle (x1,…,x n),
f kx kx kx kf x x xn n(,,,) (,,,)1 2 1 2? ??
then the technology described by the
production function f exhibits constant
returns-to-scale（ 规模报酬不变）,
E.g,(k = 2) doubling all input levels
doubles the output level.
Returns-to-Scale
y = f(x)
x? x
Input Level
Output Level
y?
One input,one output
2x?
2y?
Constant
returns-to-scale
Returns-to-Scale
If,for any input bundle (x1,…,x n),
f kx kx kx kf x x xn n(,,,) (,,,)1 2 1 2? ??
then the technology exhibits diminishing
returns-to-scale （ 规模报酬递减）,
E.g,(k = 2) doubling all input levels less
than doubles the output level.
Returns-to-Scale
y = f(x)
x? x
Input Level
Output Level
f(x?)
One input,one output
2x?
f(2x?)
2f(x?)
Decreasing
returns-to-scale
Returns-to-Scale
If,for any input bundle (x1,…,x n),
f kx kx kx kf x x xn n(,,,) (,,,)1 2 1 2? ??
then the technology exhibits increasing
returns-to-scale （ 规模报酬递增）,
E.g,(k = 2) doubling all input levels
more than doubles the output level.
Returns-to-Scale
y = f(x)
x? x
Input Level
Output Level
f(x?)
One input,one output
2x?
f(2x?)
2f(x?)
Increasing
returns-to-scale
Returns-to-Scale
?A single technology can ?locally?
exhibit different returns-to-scale.
Returns-to-Scale
y = f(x)
x
Input Level
Output Level
One input,one output
Decreasing
returns-to-scale
Increasing
returns-to-scale
Examples of Returns-to-Scale
y a x a x a xn n? ? ? ?1 1 2 2 ?,
The perfect-substitutes production
function is
Expand all input levels proportionately
by k,The output level becomes
a kx a kx a kx
k a x a x a x
ky
n n
n n
1 1 2 2
1 1 2 2
( ) ( ) ( )
( )
.
? ? ?
? ? ? ?
?
?
?
The perfect-substitutes production
function exhibits constant returns-to-scale.
Examples of Returns-to-Scale
y a x a x a xn n? m i n {,,,}.1 1 2 2 ?
The perfect-complements production
function is
Expand all input levels proportionately
by k,The output level becomes
m in{ ( ),( ),,( )}
( m in{,,,})
.
a kx a kx a kx
k a x a x a x
ky
n n
n n
1 1 2 2
1 1 2 2
?
??
?
The perfect-complements production
function exhibits constant returns-to-scale.
Examples of Returns-to-Scale
y x x xa a na n? 1 21 2 ?,
The Cobb-Douglas production function is
Expand all input levels proportionately
by k,The output level becomes
( ) ( ) ( )
.
kx kx kx
k k k x x x
k x x x
k y
a a
n
a
a a a a a a
a a a a a
n
a
a a
n
n n
n n
n
1 2
1 2
1 2
1 2 1 2
1 2 1 2
1
?
? ?
?
?
?
?
?
?
? ? ?
? ?
Examples of Returns-to-Scale
y x x xa a na n? 1 21 2 ?,
The Cobb-Douglas production function is
( ) ( ) ( ),kx kx kx k ya a n a a an n1 21 2 1? ?? ? ?
The Cobb-Douglas technology?s returns-
to-scale is
constant if a1+ … + a n = 1
increasing if a1+ … + a n > 1
decreasing if a1+ … + a n < 1.
Returns-to-Scale
?Q,Can a technology exhibit
increasing returns-to-scale even
though all of its marginal products
are diminishing?
Returns-to-Scale
?Q,Can a technology exhibit
increasing returns-to-scale even if all
of its marginal products are
diminishing?
?A,Yes.
?E.g,y x x? 12 3 22 3/ /,
Returns-to-Scale
y x x x xa a? ?12 3 22 3 1 21 2/ /
a a1 2 43 1? ? ?so this technology exhibits
increasing returns-to-scale.
But MP x x1 1
1/ 3 22 32
3?
? /
diminishes as x1
increases and
MP x x2 12 3 2 1/ 323? ?/diminishes as x
1
increases.
Returns-to-Scale
?A marginal product is the rate-of-
change of output as one input level
increases,holding all other input
levels fixed.
?Marginal product diminishes
because the other input levels are
fixed,so the increasing input?s units
have each less and less of other
inputs with which to work.
Returns-to-Scale
?When all input levels are increased
proportionately,there need be no
diminution of marginal products
since each input will always have the
same amount of other inputs with
which to work,Input productivities
need not fall and so returns-to-scale
can be constant or increasing.
Technical Rate-of-Substitution
(技术替代率 )
?At what rate can a firm substitute one
input for another without changing
its output level?
Technical Rate-of-Substitution
x2
x1
y????
x2'
x1'
Technical Rate-of-Substitution
x2
x1
y????
The slope is the rate at which
input 2 must be given up as
input 1?s level is increased so as
not to change the output level,
The slope of an isoquant is its
technical rate-of-substitution.x2'
x1'
Technical Rate-of-Substitution
?How is a technical rate-of-substitution
computed?
?The production function is
?A small change (dx1,dx2) in the input
bundle causes a change to the output
level of
y f x x? (,).1 2
dy y
x
dx y
x
dx? ??
?
?
?1 1 2 2
.
Technical Rate-of-Substitution
dy y
x
dx y
x
dx? ??
?
?
?1 1 2 2
.
But dy = 0 since there is to be no change
to the output level,so the changes dx1
and dx2 to the input levels must satisfy
0
1
1
2
2? ?
?
?
?
?
y
x
dx y
x
dx,
Technical Rate-of-Substitution
0
1
1
2
2? ?
?
?
?
?
y
x
dx y
x
dx
rearranges to?
?
?
?
y
x
dx y
x
dx
2
2
1
1? ?
so dx
dx
y x
y x
2
1
1
2
? ? ? ?
? ?
/
/
.
Technical Rate-of-Substitutiondx
dx
y x
y x
2
1
1
2
? ? ? ?? ?//
is the rate at which input 2 must be given
up as input 1 increases so as to keep
the output level constant,It is the slope
of the isoquant.
Technical Rate-of-Substitution; A
Cobb-Douglas Example
y f x x x xa b? ?(,)1 2 1 2
so ?
?
y
x
ax xa b
1
1
1
2?
??
?
y
x
bx xa b
2
1 2
1? ?,
and
The technical rate-of-substitution isdx
dx
y x
y x
ax x
bx x
ax
bx
a b
a b
2
1
1
2
1
1
2
1 2
1
2
1
? ? ? ? ? ?
?
?
? ?
? ?
/
/
.
x2
x1
Technical Rate-of-Substitution; A
Cobb-Douglas Example
T R S axbx xx x x? ? ? ? ? ?2
1
2
1
2
1
1 3
2 3 2
( / )
( / )
y x x a a n d b? ? ?11/ 3 22 3 13 23/ ;
x2
x1
Technical Rate-of-Substitution; A
Cobb-Douglas Example
T R S axbx xx x x? ? ? ? ? ?2
1
2
1
2
1
1 3
2 3 2
( / )
( / )
y x x a a n d b? ? ?11/ 3 22 3 13 23/ ;
8
4
T R S x x? ? ? ? ? ? ?2
12
8
2 4 1
x2
x1
Technical Rate-of-Substitution; A
Cobb-Douglas Example
T R S axbx xx x x? ? ? ? ? ?2
1
2
1
2
1
1 3
2 3 2
( / )
( / )
y x x a a n d b? ? ?11/ 3 22 3 13 23/ ;
6
12
T R S x x? ? ? ? ? ? ?2
12
6
2 12
1
4
Well-Behaved Technologies
?A well-behaved technology is
?monotonic,and
?convex.
Well-Behaved Technologies -
Monotonicity
?Monotonicity,More of any input
generates more output.
y
x
y
x
monotonic
not
monotonic
Well-Behaved Technologies -
Convexity
?Convexity,If the input bundles x?
and x” both provide y units of output
then the mixture tx? + (1-t)x”
provides at least y units of output,
for any 0 < t < 1,
Well-Behaved Technologies -
Convexity
x2
x1
x2'
x1'
x2"
x1"
y????
Well-Behaved Technologies -
Convexity
x2
x1
x2'
x1'
x2"
x1"
? ?tx t x tx t x1 1 2 21 1' " ' "( ),( )? ? ? ?
y????
Well-Behaved Technologies -
Convexity
x2
x1
x2'
x1'
x2"
x1"
? ?tx t x tx t x1 1 2 21 1' " ' "( ),( )? ? ? ?
y????
y????
Well-Behaved Technologies -
Convexity
x2
x1
x2'
x1'
x2"
x1"
Convexity implies that the TRS
increases (becomes less
negative) as x1 increases.
Well-Behaved Technologies –
Monotonicity and convexity
x2
x1
y????y???
y????
higher output
The Long-Run and the Short-Runs
?The long-run is the circumstance in
which a firm is unrestricted in its
choice of all input levels.
?There are many possible short-runs.
?A short-run is a circumstance in
which a firm is restricted in some
way in its choice of at least one input
level.
The Long-Run and the Short-Runs
?Examples of restrictions that place a
firm into a short-run:
?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
?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.
The Long-Run and the Short-Runs
x1
y
Four short-run production functions.
The Long-Run and the Short-Runs
y x x? 11/ 3 21/ 3is the long-run production
function (both x1 and x2 are variable).
The short-run production function when
x2 ? 1 is,x1xy 3/113/13/11 ??
The short-run production function when
x2 ? 10 is,x15210xy 3/113/13/11 ???
The Long-Run and the Short-Runs
x1
y
Four short-run production functions.
3/13/11 10xy ?
3/13/11 5xy ?
3/13/11 2xy ?
3/13/11 1xy ?
Technology
技术
Structure
? Describing technologies
?Production set or technology set
?Production function
?Isoquant
? Marginal product
? Returns to scale
? Technical rate of substitution
? Well-behaved technologies
? Long run and short run
Technologies
?A technology is a process by which
inputs are converted to an output.
?E.g,labor,a computer,a projector,
electricity,and software are being
combined to produce this lecture.
Technologies
?Usually several technologies will
produce the same product -- a
blackboard and chalk can be used
instead of a computer and a
projector.
?Which technology is,best”?
?How do we compare technologies?
Input Bundles
?xi denotes the amount used of input i;
i.e,the level of input i.
?An input bundle is a vector of the
input levels; (x1,x2,…,x n).
Production Functions (生产函数）
?y denotes the output level.
?The technology?s production
function states the maximum amount
of output possible from an input
bundle.
y f x x n? (,,)1 ?
Production Functions
y = f(x) is the
production
function.
x? x
Input Level
Output Level
y?
y? = f(x?) is the maximal
output level obtainable
from x? input units.
One input,one output
Technology Sets
? A production plan is an input bundle and an
output level; (x1,…,x n,y).
? A production plan is feasible （ 可行） if
? The collection of all feasible production
plans is the production set (生产集 ) or
technology set （ 技术集）,
y f x x n? (,,)1 ?
Technology Sets
y = f(x) is the
production
function.
x? x
Input Level
Output Level
y?
y”
y? = f(x?) is the maximal
output level obtainable
from x? input units.
One input,one output
y” = f(x?) is an output level
that is feasible from x?
input units.
Technology Sets
The production set or technology set is T x x y y f x x a n d
x x
n n
n
? ?
? ?
{(,,,) | (,,)
,,}.
1 1
1 0 0
? ?
?
Technology Sets
x? x
Input Level
Output Level
y?
One input,one output
y”
The technology
set
Technology Sets
x? x
Input Level
Output Level
y?
One input,one output
y”
The technology
setTechnically
inefficient
plans
Technically
efficient plans
Technologies with Multiple Inputs
?What does a technology look like
when there is more than one input?
?The two input case,Input levels are
x1 and x2,Output level is y.
?Suppose the production function is
y f x x x x? ?(,),1 2 11/ 3 21/ 32
Technologies with Multiple Inputs
?E.g,the maximal output level
possible from the input bundle
(x1,x2) = (1,8) is
?And the maximal output level
possible from (x1,x2) = (8,8) is
y x x? ? ? ? ? ? ? ?2 2 1 8 2 1 2 411/ 3 21/ 3 1/ 3 1/ 3,
y x x? ? ? ? ? ? ? ?2 2 8 8 2 2 2 811/ 3 21/ 3 1/ 3 1/ 3,
Technologies with Multiple Inputs
?An isoquant (等产量线 ) is the set of
all possible combinations of inputs 1
and 2 that are just sufficient to
produce a given amount of output.
Isoquants with Two Variable Inputs
y ???
y ???
x1
x2
Technologies with Multiple Inputs
?The complete collection of isoquants
is the isoquant map.
?The isoquant map is equivalent to
the production function -- each is the
other.
?E.g,3/123/1121 2),( xxxxfy ??
Isoquants with Two Variable Inputs
y ???
y ???
x1
x2
y ???
y ???
Cobb-Douglas Technologies
?A Cobb-Douglas production function
is of the form
?E.g.
with
y A x x xa a na n? ? ?1 21 2 ?,
y x x? 11/ 3 21/ 3
n A a a n d a? ? ? ?2 1 13 131 2,,.
x2
x1
All isoquants are hyperbolic
(双曲线 ),asymptoting (渐进 ) to,
but never touching any axis.
Cobb-Douglas Technologies
y x xa a? 1 21 2
x2
x1
All isoquants are hyperbolic,
asymptoting to,but never
touching any axis.
Cobb-Douglas Technologies
x x ya a1 21 2 ? '
x x ya a1 21 2 ? "
y" y'> y x x
a a? 1 21 2
Fixed-Proportions Technologies
x2
x1
min{x1,2x2} = 14
4 8 14
24
7
min{x1,2x2} = 8
min{x1,2x2} = 4
x1 = 2x2
y x x? m i n {,}1 22
Perfect-Substitution Technologies
9
3
18
6
24
8
x1
x2
x1 + 3x2 = 18
x1 + 3x2 = 36
x1 + 3x2 = 48
All are linear and parallel
y x x? ?1 23
Marginal (Physical) Products
?The marginal product (边际产量 ) of
input i is the rate-of-change of the
output level as the level of input i
changes,holding all other input
levels fixed.
?That is,
y f x x n? (,,)1 ?
i
i x
y
MP
?
?
?
Marginal (Physical) Products
E.g,if
y f x x x x? ?(,) /1 2 11/ 3 22 3
then the marginal product of input 1 is
MP yx x x1
1
1
2 3
2
2 31
3? ?
??
?
/ /
and the marginal product of input 2 isMP y
x x x2 2 1
1/ 3
2
1/ 32
3? ?
??
?,
Marginal (Physical) Products
Typically the marginal product of one
input depends upon the amount used of
other inputs,E.g,if
MP x x1 1 2 3 22 313? ? / /then,
MP x x1 1 2 3 2 3 1 2 313 8 43? ?? ?/ / /
and if x2 = 27 then
if x2 = 8,
MP x x1 1 2 3 2 3 1 2 313 27 3? ?? ?/ / /,
Marginal (Physical) Products
?The marginal product of input i is
diminishing if it becomes smaller as
the level of input i increases,That is,
if
.02
2
????
?
?
???
?
?
iiii
i
x
y
x
y
xx
MP
?
?
?
?
?
?
?
?
Marginal (Physical) Products
MP x x1 1 2 3 22 313? ? / /MP x2 11/ 3 2 1/ 323? ?and
so ?
?
MP
x x x
1
1
1
5 3
2
2 32
9 0? ? ?
? / /
?
?
MP
x x x
2
2
11/ 3 2 4 3
2
9 0? ? ?
? /,and
Both marginal products are diminishing.
E.g,if y x x? 11/ 3 22 3/then
Returns-to-Scale（ 规模收益）
?Marginal products describe the
change in output level as a single
input level changes.
?Returns-to-scale describes how the
output level changes as all input
levels change in direct proportion
(e.g,all input levels doubled,or
halved).
Returns-to-Scale
If,for any input bundle (x1,…,x n),
f kx kx kx kf x x xn n(,,,) (,,,)1 2 1 2? ??
then the technology described by the
production function f exhibits constant
returns-to-scale（ 规模报酬不变）,
E.g,(k = 2) doubling all input levels
doubles the output level.
Returns-to-Scale
y = f(x)
x? x
Input Level
Output Level
y?
One input,one output
2x?
2y?
Constant
returns-to-scale
Returns-to-Scale
If,for any input bundle (x1,…,x n),
f kx kx kx kf x x xn n(,,,) (,,,)1 2 1 2? ??
then the technology exhibits diminishing
returns-to-scale （ 规模报酬递减）,
E.g,(k = 2) doubling all input levels less
than doubles the output level.
Returns-to-Scale
y = f(x)
x? x
Input Level
Output Level
f(x?)
One input,one output
2x?
f(2x?)
2f(x?)
Decreasing
returns-to-scale
Returns-to-Scale
If,for any input bundle (x1,…,x n),
f kx kx kx kf x x xn n(,,,) (,,,)1 2 1 2? ??
then the technology exhibits increasing
returns-to-scale （ 规模报酬递增）,
E.g,(k = 2) doubling all input levels
more than doubles the output level.
Returns-to-Scale
y = f(x)
x? x
Input Level
Output Level
f(x?)
One input,one output
2x?
f(2x?)
2f(x?)
Increasing
returns-to-scale
Returns-to-Scale
?A single technology can ?locally?
exhibit different returns-to-scale.
Returns-to-Scale
y = f(x)
x
Input Level
Output Level
One input,one output
Decreasing
returns-to-scale
Increasing
returns-to-scale
Examples of Returns-to-Scale
y a x a x a xn n? ? ? ?1 1 2 2 ?,
The perfect-substitutes production
function is
Expand all input levels proportionately
by k,The output level becomes
a kx a kx a kx
k a x a x a x
ky
n n
n n
1 1 2 2
1 1 2 2
( ) ( ) ( )
( )
.
? ? ?
? ? ? ?
?
?
?
The perfect-substitutes production
function exhibits constant returns-to-scale.
Examples of Returns-to-Scale
y a x a x a xn n? m i n {,,,}.1 1 2 2 ?
The perfect-complements production
function is
Expand all input levels proportionately
by k,The output level becomes
m in{ ( ),( ),,( )}
( m in{,,,})
.
a kx a kx a kx
k a x a x a x
ky
n n
n n
1 1 2 2
1 1 2 2
?
??
?
The perfect-complements production
function exhibits constant returns-to-scale.
Examples of Returns-to-Scale
y x x xa a na n? 1 21 2 ?,
The Cobb-Douglas production function is
Expand all input levels proportionately
by k,The output level becomes
( ) ( ) ( )
.
kx kx kx
k k k x x x
k x x x
k y
a a
n
a
a a a a a a
a a a a a
n
a
a a
n
n n
n n
n
1 2
1 2
1 2
1 2 1 2
1 2 1 2
1
?
? ?
?
?
?
?
?
?
? ? ?
? ?
Examples of Returns-to-Scale
y x x xa a na n? 1 21 2 ?,
The Cobb-Douglas production function is
( ) ( ) ( ),kx kx kx k ya a n a a an n1 21 2 1? ?? ? ?
The Cobb-Douglas technology?s returns-
to-scale is
constant if a1+ … + a n = 1
increasing if a1+ … + a n > 1
decreasing if a1+ … + a n < 1.
Returns-to-Scale
?Q,Can a technology exhibit
increasing returns-to-scale even
though all of its marginal products
are diminishing?
Returns-to-Scale
?Q,Can a technology exhibit
increasing returns-to-scale even if all
of its marginal products are
diminishing?
?A,Yes.
?E.g,y x x? 12 3 22 3/ /,
Returns-to-Scale
y x x x xa a? ?12 3 22 3 1 21 2/ /
a a1 2 43 1? ? ?so this technology exhibits
increasing returns-to-scale.
But MP x x1 1
1/ 3 22 32
3?
? /
diminishes as x1
increases and
MP x x2 12 3 2 1/ 323? ?/diminishes as x
1
increases.
Returns-to-Scale
?A marginal product is the rate-of-
change of output as one input level
increases,holding all other input
levels fixed.
?Marginal product diminishes
because the other input levels are
fixed,so the increasing input?s units
have each less and less of other
inputs with which to work.
Returns-to-Scale
?When all input levels are increased
proportionately,there need be no
diminution of marginal products
since each input will always have the
same amount of other inputs with
which to work,Input productivities
need not fall and so returns-to-scale
can be constant or increasing.
Technical Rate-of-Substitution
(技术替代率 )
?At what rate can a firm substitute one
input for another without changing
its output level?
Technical Rate-of-Substitution
x2
x1
y????
x2'
x1'
Technical Rate-of-Substitution
x2
x1
y????
The slope is the rate at which
input 2 must be given up as
input 1?s level is increased so as
not to change the output level,
The slope of an isoquant is its
technical rate-of-substitution.x2'
x1'
Technical Rate-of-Substitution
?How is a technical rate-of-substitution
computed?
?The production function is
?A small change (dx1,dx2) in the input
bundle causes a change to the output
level of
y f x x? (,).1 2
dy y
x
dx y
x
dx? ??
?
?
?1 1 2 2
.
Technical Rate-of-Substitution
dy y
x
dx y
x
dx? ??
?
?
?1 1 2 2
.
But dy = 0 since there is to be no change
to the output level,so the changes dx1
and dx2 to the input levels must satisfy
0
1
1
2
2? ?
?
?
?
?
y
x
dx y
x
dx,
Technical Rate-of-Substitution
0
1
1
2
2? ?
?
?
?
?
y
x
dx y
x
dx
rearranges to?
?
?
?
y
x
dx y
x
dx
2
2
1
1? ?
so dx
dx
y x
y x
2
1
1
2
? ? ? ?
? ?
/
/
.
Technical Rate-of-Substitutiondx
dx
y x
y x
2
1
1
2
? ? ? ?? ?//
is the rate at which input 2 must be given
up as input 1 increases so as to keep
the output level constant,It is the slope
of the isoquant.
Technical Rate-of-Substitution; A
Cobb-Douglas Example
y f x x x xa b? ?(,)1 2 1 2
so ?
?
y
x
ax xa b
1
1
1
2?
??
?
y
x
bx xa b
2
1 2
1? ?,
and
The technical rate-of-substitution isdx
dx
y x
y x
ax x
bx x
ax
bx
a b
a b
2
1
1
2
1
1
2
1 2
1
2
1
? ? ? ? ? ?
?
?
? ?
? ?
/
/
.
x2
x1
Technical Rate-of-Substitution; A
Cobb-Douglas Example
T R S axbx xx x x? ? ? ? ? ?2
1
2
1
2
1
1 3
2 3 2
( / )
( / )
y x x a a n d b? ? ?11/ 3 22 3 13 23/ ;
x2
x1
Technical Rate-of-Substitution; A
Cobb-Douglas Example
T R S axbx xx x x? ? ? ? ? ?2
1
2
1
2
1
1 3
2 3 2
( / )
( / )
y x x a a n d b? ? ?11/ 3 22 3 13 23/ ;
8
4
T R S x x? ? ? ? ? ? ?2
12
8
2 4 1
x2
x1
Technical Rate-of-Substitution; A
Cobb-Douglas Example
T R S axbx xx x x? ? ? ? ? ?2
1
2
1
2
1
1 3
2 3 2
( / )
( / )
y x x a a n d b? ? ?11/ 3 22 3 13 23/ ;
6
12
T R S x x? ? ? ? ? ? ?2
12
6
2 12
1
4
Well-Behaved Technologies
?A well-behaved technology is
?monotonic,and
?convex.
Well-Behaved Technologies -
Monotonicity
?Monotonicity,More of any input
generates more output.
y
x
y
x
monotonic
not
monotonic
Well-Behaved Technologies -
Convexity
?Convexity,If the input bundles x?
and x” both provide y units of output
then the mixture tx? + (1-t)x”
provides at least y units of output,
for any 0 < t < 1,
Well-Behaved Technologies -
Convexity
x2
x1
x2'
x1'
x2"
x1"
y????
Well-Behaved Technologies -
Convexity
x2
x1
x2'
x1'
x2"
x1"
? ?tx t x tx t x1 1 2 21 1' " ' "( ),( )? ? ? ?
y????
Well-Behaved Technologies -
Convexity
x2
x1
x2'
x1'
x2"
x1"
? ?tx t x tx t x1 1 2 21 1' " ' "( ),( )? ? ? ?
y????
y????
Well-Behaved Technologies -
Convexity
x2
x1
x2'
x1'
x2"
x1"
Convexity implies that the TRS
increases (becomes less
negative) as x1 increases.
Well-Behaved Technologies –
Monotonicity and convexity
x2
x1
y????y???
y????
higher output
The Long-Run and the Short-Runs
?The long-run is the circumstance in
which a firm is unrestricted in its
choice of all input levels.
?There are many possible short-runs.
?A short-run is a circumstance in
which a firm is restricted in some
way in its choice of at least one input
level.
The Long-Run and the Short-Runs
?Examples of restrictions that place a
firm into a short-run:
?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
?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.
The Long-Run and the Short-Runs
x1
y
Four short-run production functions.
The Long-Run and the Short-Runs
y x x? 11/ 3 21/ 3is the long-run production
function (both x1 and x2 are variable).
The short-run production function when
x2 ? 1 is,x1xy 3/113/13/11 ??
The short-run production function when
x2 ? 10 is,x15210xy 3/113/13/11 ???
The Long-Run and the Short-Runs
x1
y
Four short-run production functions.
3/13/11 10xy ?
3/13/11 5xy ?
3/13/11 2xy ?
3/13/11 1xy ?
