Download - Ch18
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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. labor, 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 used instead of a computer and a projector.
Which technology is “best”?How do we compare technologies?
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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, … , xn).
E.g. (x1, x2, x3) = (6, 0, 93).
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Production Functions
y denotes the output level.The technology’s production
function states the maximum amount of output possible from an input bundle.
y f x xn ( , , )1
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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
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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
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Technology Sets
y = f(x) is theproductionfunction.
x’ xInput 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.
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Technology Sets
The technology set is
T x x y y f x x and
x xn n
n
{( , , , ) | ( , , )
, , }.1 1
1 0 0
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Technology Sets
x’ xInput Level
Output Level
y’
One input, one output
y”
The technologyset
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Technology Sets
x’ xInput Level
Output Level
y’
One input, one output
y”
The technologysetTechnically
inefficientplans
Technicallyefficient 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 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
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Technologies with Multiple 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 .
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Technologies with Multiple Inputs
Output, y
x1
x2
(8,1)(8,8)
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Technologies with Multiple Inputs
The y output unit isoquant is the set of all input bundles that yield at most the same output level y.
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Isoquants with Two Variable Inputs
y
y x1
x2
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Isoquants with Two Variable Inputs
Isoquants can be graphed by adding an output level axis and displaying each isoquant at the height of the isoquant’s output level.
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Isoquants with Two Variable Inputs
Output, y
x1
x2
y
y
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Isoquants with Two Variable Inputs
More isoquants tell us more about the technology.
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Isoquants with Two Variable Inputs
y
y
x1
x2
y
y
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Isoquants with Two Variable Inputs
Output, y
x1
x2
y
y
y
y
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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. 3/12
3/1121 2),( xxxxfy
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Technologies with Multiple Inputs
x1
x2
y
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Technologies with Multiple Inputs
x1
x2
y
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Technologies with Multiple Inputs
x1
x2
y
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Technologies with Multiple Inputs
x1
x2
y
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Technologies with Multiple Inputs
x1
x2
y
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Technologies with Multiple Inputs
x1
x2
y
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Technologies with Multiple Inputs
x1
y
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Technologies with Multiple Inputs
x1
y
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Technologies with Multiple Inputs
x1
y
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Technologies with Multiple Inputs
x1
y
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Technologies with Multiple Inputs
x1
y
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Technologies with Multiple Inputs
x1
y
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Technologies with Multiple Inputs
x1
y
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Technologies with Multiple Inputs
x1
y
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Technologies with Multiple Inputs
x1
y
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Technologies with Multiple Inputs
x1
y
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Cobb-Douglas Technologies
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, , .
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x2
x1
All isoquants are hyperbolic,asymptoting to, but nevertouching any axis.
Cobb-Douglas Technologies
y x xa a 1 21 2
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x2
x1
All isoquants are hyperbolic,asymptoting to, but nevertouching any axis.
Cobb-Douglas Technologies
x x ya a1 21 2 "
y x xa a 1 21 2
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x2
x1
All isoquants are hyperbolic,asymptoting to, but nevertouching any axis.
Cobb-Douglas Technologies
x x ya a1 21 2 '
x x ya a1 21 2 "
y x xa a 1 21 2
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x2
x1
All isoquants are hyperbolic,asymptoting to, but nevertouching any axis.
Cobb-Douglas Technologies
x x ya a1 21 2 '
x x ya a1 21 2 "
y" y'>y x xa a 1 2
1 2
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Fixed-Proportions Technologies
A fixed-proportions production function is of the form
E.g.
with
y a x a x a xn nmin{ , , , }.1 1 2 2
y x xmin{ , }1 22
n a and a 2 1 21 2, .
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Fixed-Proportions Technologies
x2
x1
min{x1,2x2} = 14
4 8 14
247
min{x1,2x2} = 8min{x1,2x2} = 4
x1 = 2x2
y x xmin{ , }1 22
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Perfect-Substitutes Technologies
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, .
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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
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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, holding all other input levels fixed.
That is,
y f x xn ( , , )1
ii x
yMP
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Marginal (Physical) Products
E.g. ify f x x x x ( , ) /
1 2 11/3
22 3
then the marginal product of input 1 is
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Marginal (Physical) Products
E.g. ify f x x x x ( , ) /
1 2 11/3
22 3
then the marginal product of input 1 is
MPyx
x x11
12 3
22 31
3
/ /
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Marginal (Physical) Products
E.g. ify f x x x x ( , ) /
1 2 11/3
22 3
then the marginal product of input 1 is
MPyx
x x11
12 3
22 31
3
/ /
and the marginal product of input 2 is
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Marginal (Physical) Products
E.g. ify f x x x x ( , ) /
1 2 11/3
22 3
then the marginal product of input 1 is
MPyx
x x11
12 3
22 31
3
/ /
and the marginal product of input 2 is
MPy
xx x2
211/3
21/32
3
.
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Marginal (Physical) Products
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 / / / .
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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
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Marginal (Physical) Products
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
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Marginal (Physical) Products
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
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Marginal (Physical) Products
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
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Marginal (Physical) Products
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
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Returns-to-Scale
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).
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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.
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Returns-to-Scale
y = f(x)
x’ xInput Level
Output Level
y’
One input, one output
2x’
2y’
Constantreturns-to-scale
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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 exhibits diminishingreturns-to-scale.E.g. (k = 2) doubling all input levels less than doubles the output level.
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Returns-to-Scale
y = f(x)
x’ xInput Level
Output Level
f(x’)
One input, one output
2x’
f(2x’)
2f(x’)
Decreasingreturns-to-scale
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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 exhibits increasingreturns-to-scale.E.g. (k = 2) doubling all input levelsmore than doubles the output level.
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Returns-to-Scale
y = f(x)
x’ xInput Level
Output Level
f(x’)
One input, one output
2x’
f(2x’)
2f(x’)
Increasingreturns-to-scale
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Returns-to-Scale
A single technology can ‘locally’ exhibit different returns-to-scale.
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Returns-to-Scale
y = f(x)
xInput Level
Output Level
One input, one output
Decreasingreturns-to-scale
Increasingreturns-to-scale
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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 kxn n1 1 2 2( ) ( ) ( )
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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 xn n
n n
1 1 2 2
1 1 2 2
( ) ( ) ( )
( )
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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.
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Examples of Returns-to-Scale
y a x a x a xn nmin{ , , , }.1 1 2 2
The perfect-complements productionfunction is
Expand all input levels proportionatelyby k. The output level becomes
min{ ( ), ( ), , ( )}a kx a kx a kxn n1 1 2 2
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Examples of Returns-to-Scale
y a x a x a xn nmin{ , , , }.1 1 2 2
The perfect-complements productionfunction is
Expand all input levels proportionatelyby k. The output level becomes
min{ ( ), ( ), , ( )}
(min{ , , , })
a kx a kx a kx
k a x a x a xn n
n n
1 1 2 2
1 1 2 2
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Examples of Returns-to-Scale
y a x a x a xn nmin{ , , , }.1 1 2 2
The perfect-complements productionfunction is
Expand all input levels proportionatelyby k. The output level becomes
min{ ( ), ( ), , ( )}
(min{ , , , })
.
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 productionfunction exhibits constant returns-to-scale.
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Examples of Returns-to-Scale
y x x xa anan 1 2
1 2 .The Cobb-Douglas production function is
Expand all input levels proportionatelyby k. The output level becomes
( ) ( ) ( )kx kx kxa an
an1 2
1 2
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Examples of Returns-to-Scale
y x x xa anan 1 2
1 2 .The Cobb-Douglas production function is
Expand all input levels proportionatelyby k. The output level becomes
( ) ( ) ( )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
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Examples of Returns-to-Scale
y x x xa anan 1 2
1 2 .The Cobb-Douglas production function is
Expand all input levels proportionatelyby k. The output level becomes
( ) ( ) ( )kx kx kx
k k k x x x
k x x x
a an
a
a a a a a a
a a a a ana
n
n n
n n
1 2
1 2
1 2
1 2 1 2
1 2 1 2
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Examples of Returns-to-Scale
y x x xa anan 1 2
1 2 .The Cobb-Douglas production function is
Expand all input levels proportionatelyby k. The output level becomes
( ) ( ) ( )
.
kx kx kx
k k k x x x
k x x x
k y
a an
a
a a a a a a
a a a a ana
a a
n
n n
n n
n
1 2
1 2
1 2
1 2 1 2
1 2 1 2
1
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Examples of Returns-to-Scale
y x x xa anan 1 2
1 2 .The Cobb-Douglas production function is
( ) ( ) ( ) .kx kx kx k ya an
a a an n1 2
1 2 1
The Cobb-Douglas technology’s returns-to-scale isconstant if a1+ … + an = 1
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Examples of Returns-to-Scale
y x x xa anan 1 2
1 2 .The Cobb-Douglas production function is
( ) ( ) ( ) .kx kx kx k ya an
a a an n1 2
1 2 1
The Cobb-Douglas technology’s returns-to-scale isconstant if a1+ … + an = 1increasing if a1+ … + an > 1
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Examples of Returns-to-Scale
y x x xa anan 1 2
1 2 .The Cobb-Douglas production function is
( ) ( ) ( ) .kx kx kx k ya an
a a an n1 2
1 2 1
The Cobb-Douglas technology’s returns-to-scale isconstant if a1+ … + an = 1increasing if a1+ … + an > 1decreasing if a1+ … + an < 1.
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Returns-to-Scale
Q: Can a technology exhibit increasing returns-to-scale even though all of its marginal products are diminishing?
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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 1
2 322 3/ / .
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Returns-to-Scaley x x x xa a 1
2 322 3
1 21 2/ /
a a1 2431 so this technology exhibits
increasing returns-to-scale.
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Returns-to-Scaley x x x xa a 1
2 322 3
1 21 2/ /
a a1 2431 so this technology exhibits
increasing returns-to-scale.
But MP x x1 11/3
22 32
3 / diminishes as x1
increases
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Returns-to-Scaley x x x xa a 1
2 322 3
1 21 2/ /
a a1 2431 so this technology exhibits
increasing returns-to-scale.
But MP x x1 11/3
22 32
3 / diminishes as x1
increases and
MP x x2 12 3
21/32
3 / diminishes as x1
increases.
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Returns-to-Scale
So a technology can exhibit increasing returns-to-scale even if all of its marginal products are diminishing. Why?
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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.
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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.
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Technical Rate-of-Substitution
At what rate can a firm substitute one input for another without changing its output level?
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Technical Rate-of-Substitution
x2
x1
y
x2'
x1'
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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'
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Technical Rate-of-Substitution
How is a technical rate-of-substitution computed?
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Technical Rate-of-Substitution
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 .
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Technical Rate-of-Substitution
dyyx
dxy
xdx
1
12
2 .
But dy = 0 since there is to be no changeto the output level, so the changes dx1
and dx2 to the input levels must satisfy
01
12
2
yx
dxy
xdx .
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Technical Rate-of-Substitution
01
12
2
yx
dxy
xdx
rearranges to
yx
dxyx
dx2
21
1
so dxdx
y xy x
2
1
1
2
//
.
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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.
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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
//
.
![Page 98: Ch18](https://reader035.vdocuments.us/reader035/viewer/2022081603/5583a90ed8b42aff058b4904/html5/thumbnails/98.jpg)
x2
x1
Technical Rate-of-Substitution; A Cobb-Douglas Example
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
/ ;
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x2
x1
Technical Rate-of-Substitution; A Cobb-Douglas Example
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
![Page 100: Ch18](https://reader035.vdocuments.us/reader035/viewer/2022081603/5583a90ed8b42aff058b4904/html5/thumbnails/100.jpg)
x2
x1
Technical Rate-of-Substitution; A Cobb-Douglas Example
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
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Well-Behaved Technologies
A well-behaved technology ismonotonic, andconvex.
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Well-Behaved Technologies - Monotonicity
Monotonicity: More of any input generates more output.
y
x
y
x
monotonic notmonotonic
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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.
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Well-Behaved Technologies - Convexity
x2
x1
x2'
x1'
x2"
x1"
y
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Well-Behaved Technologies - Convexity
x2
x1
x2'
x1'
x2"
x1"
tx t x tx t x1 1 2 21 1' " ' "( ) , ( )
y
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Well-Behaved Technologies - Convexity
x2
x1
x2'
x1'
x2"
x1"
tx t x tx t x1 1 2 21 1' " ' "( ) , ( )
yy
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Well-Behaved Technologies - Convexity
x2
x1
x2'
x1'
x2"
x1"
Convexity implies that the TRSincreases (becomes lessnegative) as x1 increases.
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Well-Behaved Technologies
x2
x1
yy
y
higher output
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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.
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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, machinerybeing required by law to meet
affirmative action quotashaving to meet domestic content
regulations.
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The Long-Run and the Short-Runs
A useful way to think of the long-run is that the firm can choose as it pleases in which short-run circumstance to be.
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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.
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The Long-Run and the Short-Runsx2
x1y
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The Long-Run and the Short-Runs
x2
x1y
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The Long-Run and the Short-Runs
x2
x1
y
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The Long-Run and the Short-Runs
x2
x1
y
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The Long-Run and the Short-Runs
x2
x1
y
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The Long-Run and the Short-Runs
x2
x1
y
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The Long-Run and the Short-Runs
x2
x1
y
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The Long-Run and the Short-Runs
x2
x1
y
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The Long-Run and the Short-Runs
x2
x1
y
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The Long-Run and the Short-Runs
x2 x1
y
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The Long-Run and the Short-Runs
x1
y
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The Long-Run and the Short-Runs
x1
y
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The Long-Run and the Short-Runs
x1
y
Four short-run production functions.
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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
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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