ch18 technology

7/25/2019 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.


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


y!

The technology

set


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Technology Sets

$0 $

1nput 2evel

3utput 2evel

y0


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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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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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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y 8

y 4

$'

$(

y 6

y 2


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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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:.g. if

then the marginal product of input ' is


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:.g. if



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:.g. if


and the marginal product of input ( is


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:.g. if


and the marginal product of input ( is

=


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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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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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$(

$'

y1


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$(

$'

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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#ow is a technical rate-of-substitution

computed"


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#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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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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rearranges to

so


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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

=



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$(

$'


Cobb-ouglas E!a(ple



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$(

$'


Cobb-ouglas E!a(ple

6

=



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$(

$'


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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Monotonicity

@onotonicity4 @ore of anyinput

generates more output.

y

$

y

$

monotonic not

monotonic



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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 '.



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Con*e!ity

$(

$'

y1



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Con*e!ity

$(

$'

y1



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Con*e!ity

$(

$'

y1

y12



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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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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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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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$'

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


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


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 l " % t t S l


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:$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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The perfect-complements productionfunction is


E l " % t t S l


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=

E l " % t t S l


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:$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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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 l " % t t S l


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=

E l " % t t S l


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=

E l " % t t S l


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The 8obb-9ouglas technology0s returns-

to-scale is

constant if a'? ) ? an '

E!a(ples o" %eturns to Scale


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to-scale is


increasing if a'? ) ? an F '

E!a(ples o" %eturns to Scale


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to-scale is


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



But diminishes as $'

increases

%eturns to Scale


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%eturns-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.

ch18 technology

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