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Transcript of Ch18 Technology
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Chapter Eighteen
Technology
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Technologies
A technology is a process by which
inputs are converted to an output.
E.g.labour, a computer, a projector,electricity, and software are being
combined to produce this lecture.
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Technologies
Usually several technologies will
produce the same product -- a
blackboard and chalk can be usedinstead of a computer and a
projector.
Which technology is best!"
#ow do we compare technologies"
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Input Bundles
$idenotes the amount used of input i%
i.e.the level of input i.
An input bundleis a vector of theinput levels% &$', $(, ) , $n*.
E.g.&$', $(, $+* &, , /+*.
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Production Functions
y denotes the output level.
The technology0s production
functionstates the ma$imumamountof output possible from an input
bundle.
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Production Functions
y f&$* is the
production
function.
$0 $
1nput 2evel
3utput 2evel
y0 y0 f&$0* is the ma$imal
output level obtainable
from $0 input units.
3ne input, one output
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Technology Sets
A production planis an input bundle
and an output level% &$', ) , $n, y*.
A production plan is feasibleif
The collection of all feasible production
plans is the technology set.
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Technology Sets
y f&$* is the
production
function.
$0 $
1nput 2evel
3utput 2evel
y0
y!
y0 f&$0* is the ma$imaloutput level obtainablefrom $0 input units.
3ne input, one output
y! f&$0* is an output levelthat is feasible from $0input units.
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Technology Sets
The technology setis
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Technology Sets
$0 $
1nput 2evel
3utput 2evel
y0
3ne input, one output
y!
The technology
set
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Technology Sets
$0 $
1nput 2evel
3utput 2evel
y0
3ne input, one output
y!
The technology
setTechnically
inefficientplans
Technically
efficient plans
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Technologies with Multiple Inputs
What does a technology look like
when there is more than one input"
The two input case4 1nput levels are$'and $(. 3utput level is y.
5uppose the production function is
=
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Technologies with Multiple Inputs
E.g.the ma$imal output level possiblefrom the input bundle
&$', $(* &', 6* is
And the ma$imal output level possible
from &$',$(* &6,6* is
==
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Isoquants with Two Variable Inputs
y 8
y 4
$'
$(
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Isoquants with Two Variable Inputs
1so7uants can be graphed by adding
an output level a$is and displaying
each iso7uant at the height of the
iso7uant0s output level.
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Isoquants with Two Variable Inputs
y 8
y 4
$'
$(
y 6
y 2
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Technologies with Multiple Inputs
The complete collection of iso7uantsis the iso7uant map.
The iso7uant map is e7uivalent to
the production function-- each is theother.
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Cobb-ouglas Technologies
A 8obb-9ouglas production function
is of the form
:.g.
with=
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$(
$'
All iso7uants are hyperbolic,asymptoting to, but never
touching any a$is.
Cobb-ouglas Technologies
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Fi!ed-Proportions Technologies
A fi$ed-proportions production
function is of the form
:.g.
with
=
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Fi!ed-Proportions Technologies
$(
$'
min;$',($(< '=
= 6 '=
(=
>
min;$',($(< 6min;$',($(< =
$' ($(
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Per"ect-Substitutes Technologies
A perfect-substitutes production
function is of the form
:.g.
with=
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Per"ect-Substitution Technologies
/
+
'6
(=
6
$'
$(
$'? +$( '6
$'? +$( +
$'? +$( =6
All are linear and parallel
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Marginal #Physical$ Products
The marginal product of input i is the
rate-of-change of the output level as
the level of input i changes, holdingall other input levels fi$ed.
That is,
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Marginal #Physical$ Products
:.g. if
then the marginal product of input ' is
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Marginal #Physical$ Products
:.g. if
then the marginal product of input ' is
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Marginal #Physical$ Products
:.g. if
then the marginal product of input ' is
and the marginal product of input ( is
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Marginal #Physical$ Products
:.g. if
then the marginal product of input ' is
and the marginal product of input ( is
=
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Marginal #Physical$ Products
The marginal product of input i is
diminishingif it becomes smaller as
the level of input i increases. That is,
if
=
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Marginal #Physical$ Products
and
so
=
and
Both marginal products are diminishing.
:.g. if then
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Technical %ate-o"-Substitution
At what rate can a firm substitute
one input for another without
changing its output level"
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Technical %ate-o"-Substitution
$(
$'
y1
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Technical %ate-o"-Substitution
$(
$'
y1
The slope is the rate at which
input ( must be given up as
input '0s level is increased so as
not to change the output level.
The slope of an iso7uant is itstechnical rate-of-substitution.
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Technical %ate-o"-Substitution
#ow is a technical rate-of-substitution
computed"
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Technical %ate-o"-Substitution
#ow is a technical rate-of-substitution
computed"
The production function isA small change &d$', d$(* in the input
bundle causes a change to the output
level of=
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Technical %ate-o"-Substitution
But dy since there is to be no change
to the output level, so the changes d$'and d$(to the input levels must satisfy
=
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Technical %ate-o"-Substitution
rearranges to
so
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Technical %ate-o"-Substitution
is the rate at which input ( must be given
up as input ' increases so as to keepthe output level constant. 1t is the slope
of the iso7uant.
Technical %ate o" Substitution& '
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Technical %ate-o"-Substitution& '
Cobb-ouglas E!a(ple
so
and
The technical rate-of-substitution is
=
Technical %ate o" Substitution& '
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$(
$'
Technical %ate-o"-Substitution& '
Cobb-ouglas E!a(ple
Technical %ate o" Substitution& '
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$(
$'
Technical %ate-o"-Substitution& '
Cobb-ouglas E!a(ple
6
=
Technical %ate o" Substitution& '
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$(
$'
Technical %ate-o"-Substitution& '
Cobb-ouglas E!a(ple
'(
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)ell-Beha*ed Technologies
A well-behavedtechnology is
monotonic, and
conve$.
)ell-Beha*ed Technologies -
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)ell-Beha*ed Technologies -
Monotonicity
@onotonicity4 @ore of anyinput
generates more output.
y
$
y
$
monotonic not
monotonic
)ell-Beha*ed Technologies -
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)ell-Beha*ed Technologies -
Con*e!ity
8onve$ity4 1f the input bundles $0
and $! both provide y units of output
then the mi$ture t$0 ? &'-t*$!
provides at least y units of output,
for any D t D '.
)ell-Beha*ed Technologies -
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)ell-Beha*ed Technologies -
Con*e!ity
$(
$'
y1
)ell-Beha*ed Technologies -
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)ell-Beha*ed Technologies -
Con*e!ity
$(
$'
y1
)ell-Beha*ed Technologies -
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)ell-Beha*ed Technologies -
Con*e!ity
$(
$'
y1
y12
)ell-Beha*ed Technologies -
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)ell Beha*ed Technologies
Con*e!ity
$(
$'
8onve$ity implies that the TC5increases &becomes less
negative* as $'increases.
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)ell-Beha*ed Technologies
$(
$'
y1y5
y2
higher output
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The +ong-%un and the Short-%uns
The long-runis thecircumstance in
which a firm is unrestrictedin its
choice of all input levels.
There are many possible short-runs.
A short-runis acircumstance in
which a firm is restrictedin some way
in its choice of at least one input
level.
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The +ong-%un and the Short-%uns
What do short-run restrictions imply
for a firm0s technology"
5uppose the short-run restriction is
fi$ing the level of input (.
1nput ( is thus a fi$ed inputin the
short-run. 1nput ' remains variable.
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The +ong-%un and the Short-%uns
is the long-run productionfunction &both $'and $(are variable*.
The short-run production function when
$(' is
The short-run production function when
$(' is
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The +ong-%un and the Short-%uns
$'
y
Eour short-run production functions.
l
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%eturns-to-Scale
@arginal products describe the
change in output level as a single
input level changes.
Ceturns-to-scaledescribes how the
output level changes as allinput
levels change in direct proportion
&e.g.all input levels doubled, or
halved*.
S l
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%eturns-to-Scale
1f, for any input bundle &$',),$n*,
then the technology described by the
production function f e$hibits constant
returns-to-scale.
E.g. &k (* doubling all input levels
doubles the output level.
% S l
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%eturns-to-Scale
1f, for any input bundle &$',),$n*,
then the technology e$hibits diminishing
returns-to-scale.
E.g. &k (* doubling all input levels less
than doubles the output level.
% S l
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%eturns-to-Scale
1f, for any input bundle &$',),$n*,
then the technology e$hibits increasing
returns-to-scale.
E.g. &k (* doubling all input levels
more than doubles the output level.
E l " % S l
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E!a(ples o" %eturns-to-Scale
The perfect-substitutes productionfunction is
:$pand all input levels proportionatelyby k. The output level becomes
E l " % S l
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E!a(ples o" %eturns-to-Scale
The perfect-substitutes productionfunction is
:$pand all input levels proportionatelyby k. The output level becomes
=
E l " % t t S l
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E!a(ples o" %eturns-to-Scale
The perfect-substitutes productionfunction is
:$pand all input levels proportionatelyby k. The output level becomes.=The perfect-substitutes production
function e$hibits constant returns-to-scale.
E l " % t t S l
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E!a(ples o" %eturns-to-Scale
The perfect-complements productionfunction is
:$pand all input levels proportionatelyby k. The output level becomes
E l " % t t S l
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E!a(ples o" %eturns-to-Scale
The perfect-complements productionfunction is
:$pand all input levels proportionatelyby k. The output level becomes
=
E l " % t t S l
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E!a(ples o" %eturns-to-Scale
The perfect-complements productionfunction is
:$pand all input levels proportionatelyby k. The output level becomes.=The perfect-complements production
function e$hibits constant returns-to-scale.
E l " % t t S l
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E!a(ples o" %eturns-to-Scale
The 8obb-9ouglas production function is
:$pand all input levels proportionately
by k. The output level becomes
E l " % t t S l
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E!a(ples o" %eturns-to-Scale
The 8obb-9ouglas production function is
:$pand all input levels proportionately
by k. The output level becomes
=
E l " % t t S l
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E!a(ples o" %eturns-to-Scale
The 8obb-9ouglas production function is
:$pand all input levels proportionately
by k. The output level becomes
=
E l " % t t S l
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E!a(ples o" %eturns-to-Scale
The 8obb-9ouglas production function is
:$pand all input levels proportionately
by k. The output level becomes
=
E l " % t t S l
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E!a(ples o" %eturns-to-Scale
The 8obb-9ouglas production function is
The 8obb-9ouglas technology0s returns-
to-scale is
constant if a'? ) ? an '
E!a(ples o" %eturns to Scale
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E!a(ples o" %eturns-to-Scale
The 8obb-9ouglas production function is
The 8obb-9ouglas technology0s returns-
to-scale is
constant if a'? ) ? an '
increasing if a'? ) ? an F '
E!a(ples o" %eturns to Scale
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E!a(ples o" %eturns-to-Scale
The 8obb-9ouglas production function is
The 8obb-9ouglas technology0s returns-
to-scale is
constant if a'? ) ? an '
increasing if a'? ) ? an F '
decreasing if a'? ) ? an D '.
%eturns to Scale
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%eturns-to-Scale
G4 8an a technology e$hibit
increasing returns-to-scale even
though all of its marginal products
are diminishing"
%eturns to Scale
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%eturns-to-Scale
G4 8an a technology e$hibit
increasing returns-to-scale even if all
of its marginal products are
diminishing"
A4 Hes.
:.g.=
%eturns to Scale
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%eturns-to-Scale
so this technology e$hibits
increasing returns-to-scale.
%eturns to Scale
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%eturns-to-Scale
so this technology e$hibits
increasing returns-to-scale.
But diminishes as $'
increases
%eturns to Scale
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%eturns-to-Scale
so this technology e$hibits
increasing returns-to-scale.
But diminishes as $'
increases and
=
diminishes as $'increases.
%eturns to Scale
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%eturns-to-Scale
5o a technology can e$hibit
increasing returns-to-scale even if all
of its marginal products are
diminishing. Why"
%eturns to Scale
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%eturns-to-Scale
A marginal product is the rate-of-change of output as oneinput level
increases, holding all other input
levels fi$ed.
@arginal product diminishes because
the other input levels are fi$ed, so the
increasing input0s units have each less
and less of other inputs with which towork.
%eturns to Scale
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%eturns-to-Scale
When allinput levels are increasedproportionately, there need be no
diminution of marginal products
since each input will always have thesame amount of other inputs with
which to work. 1nput productivities
need not fall and so returns-to-scale
can be constant or increasing.