chapter eighteen technology. technologies a technology is a process by which inputs are converted...

Chapter Eighteen

Technology

Technologies

A technology is a process by which inputs are converted to an output

E.g. seed, chemical fertilizer, pesticides, herbicides, labour, land, machinery, fuel are combined to grow wheat

Technologies

Usually several technologies will produce the same product –

e.g. wheat can be grown with conventional technology or organically without chemicals, using animal fertilizer and more labour instead

How do economists analyse technologies?

Production Functions

y denotes the output levelxi denotes the amount used of input i; i.e. the level of input i

The technology’s production function states the maximum amount of output possible from an input bundle

y f x xn ( , , )1

Production Functions

y = f(x) is theproductionfunction.

x’ xInput 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, … , xn, y).

A production plan is feasible if

The collection of all feasible

production plans is the technology set.

y f x xn ( , , )1

Technology Sets

y = f(x) is theproductionfunction.

x’ xInput Level

Output Level

y’

y”

y’ = f(x’) is the maximum output level obtainable from x’ input units.


y” = f(x’) is an output level that is feasible from x’ input units.

Technology Sets

x’ xInput Level

Output Level

y’


y”

The technologyset

Technology Sets

x’ xInput Level

Output Level

y’


y”

The technologysetTechnically

inefficientplans

Technicallyefficient 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 Two InputsE.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 Two Inputs

Output, y

x1

x2

(8,1)(8,8)


An isoquant shows all input bundles that yield at most the same output level, say y=y1

An isoquant is drawn on a graph where the axes measure the levels of the two inputs

Isoquants with Two Variable Inputs

y

y

x1

x2

y

y

Isoquants with Two Variable Inputs

Output, y

x1

x2

y

y

y

y

Technologies with Two InputsThe complete collection of isoquants

is the isoquant mapThe isoquant map is equivalent to

the production function – they each contain the same information about the technology

E.g. 3/12

3/1121 2),( xxxxfy


x1

x2

y


x1

y

Cobb-Douglas Technology

A Cobb-Douglas production function is of the form

E.g.

with

y Ax x xa anan 1 2

1 2 .

y x x 11/3

21/3

n A a and a 2 113

131 2, , .

x2

x1

All isoquants are hyperbolic,asymptoting to, but nevertouching any axis.


y x xa a 1 21 2

x2

x1

All isoquants are hyperbolic,asymptoting to, but nevertouching any axis.


x x ya a1 21 2 '

x x ya a1 21 2 "

y" y'>y x xa a 1 2

1 2

Fixed-Proportions TechnologyA fixed-proportions production

function is of the form

E.g. y = lectures per hour in the Leeuwenborch, x1 = lecturer, x2 = lecture room

with

y a x a x a xn nmin{ , , , }.1 1 2 2

},min{ 21 xxy

1 1 ,2 21 aandan

Fixed-Proportions Technology

x2

x1

min{x1,2x2} = 14

4 8 14

247

min{x1,2x2} = 8min{x1,2x2} = 4

x1 = 2x2

y x xmin{ , }1 22

Perfect-Substitutes Technology

A perfect-substitutes production function is of the form

E.g.

with

y a x a x a xn n 1 1 2 2 .

y x x 1 23

n a and a 2 1 31 2, .

Perfect-Substitution Technology

9

3

18

6

24

8

x1

x2

x1 + 3x2 = 9

x1 + 3x2 = 18

x1 + 3x2 = 24

All are linear and parallel

y x x 1 23

Recap on Isoquants

Isoquants are generally downward- sloping from left to right (inputs are substitutes) and convex to the origin

Limiting cases: Straight line (perfect substitutes)Right angle (perfect complements)

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, holding all other input levels fixed.

That is,

y f x xn ( , , )1

ii x

yMP


E.g. ify f x x x x ( , ) /

1 2 11/3

22 3

then the marginal product of input 1 is


E.g. ify f x x x x ( , ) /

1 2 11/3

22 3


MPyx

x x11

12 3

22 31

3

/ /


E.g. ify f x x x x ( , ) /

1 2 11/3

22 3


MPyx

x x11

12 3

22 31

3

/ /

and the marginal product of input 2 is


E.g. ify f x x x x ( , ) /

1 2 11/3

22 3


MPyx

x x11

12 3

22 31

3

/ /

and the marginal product of input 2 is

MPy

xx x2

211/3

21/32

3

.


Typically the marginal product of oneinput depends upon the amount used of other inputs. E.g. if

MP x x1 12 3

22 31

3 / / then,

MP x x1 12 3 2 3

12 31

38

43

/ / /

and if x2 = 27 then

if x2 = 8,

MP x x1 12 3 2 3

12 31

327 3 / / / .


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


MP x x1 12 3

22 31

3 / / MP x x2 1

1/321/32

3

and

E.g. if y x x 11/3

22 3/ then


MP x x1 12 3

22 31

3 / / MP x x2 1

1/321/32

3

and

so MPx

x x1

115 3

22 32

90 / /

E.g. if y x x 11/3

22 3/ then


MP x x1 12 3

22 31

3 / / MP x x2 1

1/321/32

3

and

so MPx

x x1

115 3

22 32

90 / /

MPx

x x2

211/3

24 32

90 / .

and

E.g. if y x x 11/3

22 3/ then


MP x x1 12 3

22 31

3 / / MP x x2 1

1/321/32

3

and

so MPx

x x1

115 3

22 32

90 / /

MPx

x x2

211/3

24 32

90 / .

and

Both marginal products are diminishing.

E.g. if y x x 11/3

22 3/ then

Elasticity of Production

The value of the marginal physical product depends on the units of measurement of output and inputs

The elasticity of production is a unit-free measure of how the responsiveness of output to a change in an input

Elasticity of Production

Elasticity of production of input i is

i

i

ii

i

iyx

APMP

xy

xy

xx

yy

i

Elasticity of Production: Example Cobb-Douglas

For the Cobb-Douglas production function, the elasticity of production of each input is the power to which that input is raised in the function

y x xa a 1 21 2

Production elasticity of x1 is a1, production elasticity of x2 is a2

Returns to Scale

Marginal products describe the change in output level as a single input level changes

‘Returns to scale’ describe how the output level changes as all input levels change by the same proportion (e.g. all input levels doubled, or halved)

Returns to Scale

If, for any input bundle (x1,…,xn),

f kx kx kx kf x x xn n( , , , ) ( , , , )1 2 1 2

then the technology described by theproduction function f exhibits constantreturns to scale.

E.g. (k = 2) doubling all input levelsdoubles the output level.

Returns to Scale

y = f(x)

x’ xInput Level

Output Level

y’


2x’

2y’

Constantreturns to scale

Returns to Scale

),,,(),,,( 2121 nt

n xxxfkkxkxkxf

In general, for any input bundle (x1,…,xn),

If t < 1, then the technology exhibits decreasing returns to scaleE.g. (k = 2) all input levels double, but output increases by less than double

If t > 1, then the technology exhibits increasing returns to scale

),,,(),,,( nt

n xxxfkkxkxkxf 2121

Returns to Scale

y = f(x)

x’ xInput Level

Output Level

f(x’)


2x’

f(2x’)

2f(x’)

Decreasingreturns to scale

Returns to Scale

y = f(x)

x’ xInput Level

Output Level

f(x’)


2x’

f(2x’)

2f(x’)

Increasingreturns to scale

Returns to Scale

y = f(x)

xInput Level

Output Level


Decreasingreturns to scale

Increasingreturns to scale

Examples of Returns to Scale

y a x a x a xn n 1 1 2 2 .

The perfect-substitutes productionfunction is

Expand all input levels proportionatelyby 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 productionfunction exhibits constant returns to scale.


y x x xa anan 1 2

1 2 .The Cobb-Douglas production function is


( ) ( ) ( )kx kx kxa an

an1 2

1 2


y x x xa anan 1 2



( ) ( ) ( )kx kx kx

k k k x x x

a an

a

a a a a a a

n

n n

1 21 2

1 2 1 2


y x x xa anan 1 2



nn

nn

n

an

aaaaa

aaaaaa

an

aa

xxxk

xxxkkk

kxkxkx

2121

2121

21

21)(

21 )()()(


nan

aa xxxy 21

21The Cobb-Douglas production function is


yk

xxxk

xxxkkk

kxkxkx

n

nn

nn

n

aa

an

aaaaa

aaaaaa

an

aa

)(

21)(

21

1

2121

2121

21 )()()(


y x x xa anan 1 2


( ) ( ) ( ) .kx kx kx k ya an

a a an n1 2

1 2 1

The Cobb-Douglas technology’s returns to scale areconstant if a1+ … + an = 1increasing if a1+ … + an > 1decreasing if a1+ … + an < 1

Returns to Scale

Q: Can a technology exhibit increasing returns-to-scale even though all of its marginal products are diminishing?

A: Yes, because…..

Returns to Scale

A marginal product is the rate-of-change of output as one input level increases, holding all other input levels fixed, and so

the 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

BUT when all input levels are increased proportionately, there need be no diminution of marginal products since each input will always have proportionately the same amount of other inputs to work with. Input productivities need not fall and so returns-to-scale can be constant or increasing.

Technical Rate of Substitution

At what rate can a firm substitute one input for another without changing its output level?


x2

x1

y

x2'

x1'


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 substitutionx2

'

x1'


How is a technical rate of substitution computed?

The production function isA small change (dx1, dx2) in the input

bundle causes a change to the output level of

y f x x ( , ).1 2

dyyx

dxy

xdx

1

12

2 .


dyyx

dxy

xdx

1

12

2 .

But dy = 0 as we move around an isoquant, so the changes dx1and dx2 to the input levels must satisfy

01

12

2

yx

dxy

xdx .


2

1

2

1

1

2

MPMP

xyxy

dxdx

//

01

12

2

yx

dxy

xdx

rearranges to

yx

dxyx

dx2

21

1

so

2

1

2

1

1

2

//

MPMP

xyxy

dxdx

Technical Rate of Substitutiondxdx

y xy x

2

1

1

2

//

is the rate at which input 2 must be givenup as input 1 increases so as to keepthe output level constant. It is the slopeof the isoquant.

Technical Rate of Substitution; A Cobb-Douglas Example

y f x x x xa b ( , )1 2 1 2

so

yx

ax xa b

1112

y

xbx xa b

21 2

1 .and

The technical rate-of-substitution is

dxdx

y xy x

ax x

bx x

axbx

a b

a b2

1

1

2

112

1 21

2

1

//

.

Diminishing TRS when inputs are less-than-perfect substitutes

As we substitute more x1 for x2 (i.e. a move to the right around an isoquant), the TRS decreases (moves closer to zero): the slope of the isoquant becomes flatter. That is, we need ever-larger increase in x1 to substitute for a given reduction in x2, if we wish to keep output constant

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 notmonotonic

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.


x2

x1

x2'

x1'

x2"

x1"

y


x2

x1

x2'

x1'

x2"

x1"

tx t x tx t x1 1 2 21 1' " ' "( ) , ( )

y


x2

x1

x2'

x1'

x2"

x1"

tx t x tx t x1 1 2 21 1' " ' "( ) , ( )

yy


x2

x1

x2'

x1'

x2"

x1"

Convexity implies that the TRSincreases (becomes lessnegative) as x1 increases.

Well-Behaved Technologies

x2

x1

yy

y

higher output

The Long Run and the Short Run(s)

The long run is the situation in which a firm is unrestricted in its choice of all input levels

There are many possible short runs.A short run is a situation in which a

firm is restricted in some way in its choice of at least one input level

Read p.339 on fixed and variable factors

The Long Run and the Short Runs

Examples of restrictions that cause a short-run situation:temporarily being unable to

install, or remove, machinerybeing unable to increase the

number of dairy cows quicklytotal farm area is fixed in the

‘short run’


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 Runsx2

x1


x2

x1y


x2

x1

y


x2 x1

y


x1

y


x1

y

Four short-run production functions.

The Long-Run and the Short-Runs

y x x 11/3

21/3

is the long-run productionfunction (both x1 and x2 are variable).

The short-run production function whenx2 1 is .x1xy 3/1

13/13/1

1

The short-run production function when x2 10 is .x15210xy 3/1

13/13/1

1

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

x2

x1


TRSaxbx

xx

xx

2

1

2

1

2

1

1 32 3 2( / )( / )

y x x a and b 11/3

22 3 1

323

/ ;

x2

x1


TRSaxbx

xx

xx

2

1

2

1

2

1

1 32 3 2( / )( / )

y x x a and b 11/3

22 3 1

323

/ ;

8

4

TRSxx

2

1282 4

1

x2

x1


TRSaxbx

xx

xx

2

1

2

1

2

1

1 32 3 2( / )( / )

y x x a and b 11/3

22 3 1

323

/ ;

6

12

TRSxx

2

126

2 1214

chapter eighteen technology. technologies a technology is a process by which inputs are converted...

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input level output level

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