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Chapter Eighteen
Technology
Technologies
A

technology is a process by which
inputs are converted to an output.
 E.g. labor, a computer, a projector...
Technologies
 Usually

several technologies will
produce the same product -- a
blackboard and chalk can be used
instead o...
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
in...
Production Functions
y

denotes the output level.
 The technology’s production
function states the maximum amount
of out...
Production Functions
One input, one output
Output Level

y = f(x) is the
production
function.

y’

y’ = f(x’) is the maxim...
Technology Sets
A

production plan is an input bundle
and an output level; (x1, … , xn, y).

A

production plan is feasi...
Technology Sets
One input, one output
Output Level
y’

y”

y = f(x) is the
production
function.
y’ = f(x’) is the maximal
...
Technology Sets
The technology set is

T = {( x1 ,, xn , y) | y ≤ f ( x1 ,, xn ) and
x1 ≥ 0,, xn ≥ 0}.
Technology Sets
One input, one output
Output Level
y’
The technology
set

y”
x’
Input Level

x
Technology Sets
One input, one output
Output Level
Technically
efficient plans

y’

y”

The technology
Technically set
ine...
Technologies with Multiple Inputs
 What

does a technology look like
when there is more than one input?
 The two input c...
Technologies with Multiple Inputs
 E.g.

the maximal output level
possible from the input bundle
(x1, x2) = (1, 8) is

y ...
Technologies with Multiple Inputs
Output, y

x2
(8,1)
x1

(8,8)
Technologies with Multiple Inputs
 The

y output unit isoquant is the set
of all input bundles that yield at most
the sam...
Isoquants with Two Variable Inputs
x2

y≡ 8

y≡ 4
x1
Isoquants with Two Variable Inputs
 Isoquants

can be graphed by adding
an output level axis and displaying
each isoquant...
Isoquants with Two Variable Inputs
Output, y
y≡ 8

x2 y ≡ 4

x1
Isoquants with Two Variable Inputs
 More

isoquants tell us more about
the technology.
Isoquants with Two Variable Inputs
x2

y≡ 8

y≡ 6
y≡ 4
y≡ 2
x1
Isoquants with Two Variable Inputs
Output, y
y≡ 8
y≡ 6
x2 y ≡ 4
y≡ 2
x1
Technologies with Multiple Inputs
 The

complete collection of isoquants
is the isoquant map.
 The isoquant map is equiv...
Technologies with Multiple Inputs
x2

y

x1
Technologies with Multiple Inputs
x2

y
x1
Technologies with Multiple Inputs
x2

y
x1
Technologies with Multiple Inputs
x2

y
x1
Technologies with Multiple Inputs
x2

y

x1
Technologies with Multiple Inputs
x2

y

x1
Technologies with Multiple Inputs

y

x1
Technologies with Multiple Inputs

y

x1
Technologies with Multiple Inputs

y

x1
Technologies with Multiple Inputs

y

x1
Technologies with Multiple Inputs

y

x1
Technologies with Multiple Inputs

y

x1
Technologies with Multiple Inputs

y

x1
Technologies with Multiple Inputs

y

x1
Technologies with Multiple Inputs

y

x1
Technologies with Multiple Inputs

y

x1
Cobb-Douglas Technologies
A

Cobb-Douglas production function
is of the form
a1 a 2
an
y = A x1 x 2 ×× xn .

 E.g.

wit...
Cobb-Douglas Technologies

x2

All isoquants are hyperbolic,
asymptoting to, but never
touching any axis.
a1 a 2
y = x1 x ...
Cobb-Douglas Technologies

x2

All isoquants are hyperbolic,
asymptoting to, but never
touching any axis.
a1 a 2
y = x1 x ...
Cobb-Douglas Technologies

x2

All isoquants are hyperbolic,
asymptoting to, but never
touching any axis.
a1 a 2
y = x1 x ...
Cobb-Douglas Technologies

x2

All isoquants are hyperbolic,
asymptoting to, but never
touching any axis.
y" > y'

a1 a 2
...
Fixed-Proportions Technologies
A

fixed-proportions production
function is of the form
y = min{ a1 x1 , a 2x 2 ,, an xn ...
Fixed-Proportions Technologies
y = min{ x1 , 2x 2 }

x2

x1 = 2x2
7
4
2
4

8

min{x1,2x2} = 14
min{x1,2x2} = 8
min{x1,2x2}...
Perfect-Substitutes Technologies
A

perfect-substitutes production
function is of the form
y = a1 x1 + a 2x 2 +  + an xn...
Perfect-Substitution Technologies
y = x1 + 3x 2

x2

x1 + 3x2 = 18
x1 + 3x2 = 36
x1 + 3x2 = 48

8
6
3

All are linear and ...
Marginal (Physical) Products
y = f ( x 1 ,  , xn )
 The

marginal product of input i is the
rate-of-change of the output...
Marginal (Physical) Products
E.g. if
1/ 3 2 / 3
y = f ( x1 , x 2 ) = x1 x 2
then the marginal product of input 1 is
Marginal (Physical) Products
E.g. if
1/ 3 2 / 3
y = f ( x1 , x 2 ) = x1 x 2
then the marginal product of input 1 is
∂ y 1 ...
Marginal (Physical) Products
E.g. if
1/ 3 2 / 3
y = f ( x1 , x 2 ) = x1 x 2
then the marginal product of input 1 is
∂ y 1 ...
Marginal (Physical) Products
E.g. if
1/ 3 2 / 3
y = f ( x1 , x 2 ) = x1 x 2
then the marginal product of input 1 is
∂ y 1 ...
Marginal (Physical) Products
Typically the marginal product of one
input depends upon the amount used
of other inputs. E.g...
Marginal (Physical) Products
 The

marginal product of input i is
diminishing if it becomes smaller as
the level of input...
Marginal (Physical) Products
1/ 3 2 / 3
E.g. if y = x1 x 2
then
1 − 2/ 3 2/ 3
2 1/ 3 −1/ 3
MP1 = x1 x 2
and MP2 = x1 x 2

...
Marginal (Physical) Products
1/ 3 2 / 3
E.g. if y = x1 x 2
then
1 − 2/ 3 2/ 3
2 1/ 3 −1/ 3
MP1 = x1 x 2
and MP2 = x1 x 2

...
Marginal (Physical) Products
1/ 3 2 / 3
E.g. if y = x1 x 2
then
1 − 2/ 3 2/ 3
2 1/ 3 −1/ 3
MP1 = x1 x 2
and MP2 = x1 x 2

...
Marginal (Physical) Products
1/ 3 2 / 3
E.g. if y = x1 x 2
then
1 − 2/ 3 2/ 3
2 1/ 3 −1/ 3
MP1 = x1 x 2
and MP2 = x1 x 2

...
Returns-to-Scale
 Marginal

products describe the
change in output level as a single
input level changes.
 Returns-to-sc...
Returns-to-Scale
If, for any input bundle (x1,…,xn),
f (kx1 , kx 2 ,, kxn ) = kf ( x1 , x 2 ,, xn )

then the technology...
Returns-to-Scale
One input, one output
Output Level
y = f(x)

2y’

Constant
returns-to-scale

y’

x’
2x’
Input Level

x
Returns-to-Scale
If, for any input bundle (x1,…,xn),
f (kx1 , kx 2 ,, kxn ) < kf ( x1 , x 2 ,, xn )

then the technology...
Returns-to-Scale
One input, one output
Output Level
2f(x’)

y = f(x)

f(2x’)

Decreasing
returns-to-scale

f(x’)

x’

2x’
...
Returns-to-Scale
If, for any input bundle (x1,…,xn),
f (kx1 , kx 2 ,, kxn ) > kf ( x1 , x 2 ,, xn )

then the technology...
Returns-to-Scale
One input, one output
Output Level

Increasing
returns-to-scale

y = f(x)

f(2x’)
2f(x’)
f(x’)
x’
2x’
Inp...
Returns-to-Scale
A

single technology can ‘locally’
exhibit different returns-to-scale.
Returns-to-Scale
One input, one output
Output Level
y = f(x)

Increasing
returns-to-scale

Decreasing
returns-to-scale
x
I...
Examples of Returns-to-Scale
The perfect-substitutes production
function is
y = a1 x1 + a 2x 2 +  + an xn .

Expand all i...
Examples of Returns-to-Scale
The perfect-substitutes production
function is
y = a1 x1 + a 2x 2 +  + an xn .

Expand all i...
Examples of Returns-to-Scale
The perfect-substitutes production
function is
y = a1 x1 + a 2x 2 +  + an xn .

Expand all i...
Examples of Returns-to-Scale
The perfect-complements production
function is
y = min{ a1 x1 , a 2x 2 ,  , an xn }.

Expand...
Examples of Returns-to-Scale
The perfect-complements production
function is
y = min{ a1 x1 , a 2x 2 ,  , an xn }.

Expand...
Examples of Returns-to-Scale
The perfect-complements production
function is
y = min{ a1 x1 , a 2x 2 ,  , an xn }.

Expand...
Examples of Returns-to-Scale
The Cobb-Douglas production function is
a
a
y = x1 1 x a 2  xnn .
2

Expand all input levels...
Examples of Returns-to-Scale
The Cobb-Douglas production function is
a
a
y = x1 1 x a 2  xnn .
2

Expand all input levels...
Examples of Returns-to-Scale
The Cobb-Douglas production function is
a
a
y = x1 1 x a 2  xnn .
2

Expand all input levels...
Examples of Returns-to-Scale
The Cobb-Douglas production function is
a
a
y = x1 1 x a 2  xnn .
2

Expand all input levels...
Examples of Returns-to-Scale
The Cobb-Douglas production function is
a
a
y = x1 1 x a 2  xnn .
2

(kx1 ) a1 (kx 2 ) a 2 ...
Examples of Returns-to-Scale
The Cobb-Douglas production function is
a
a
y = x1 1 x a 2  xnn .
2

(kx1 ) a1 (kx 2 ) a 2 ...
Examples of Returns-to-Scale
The Cobb-Douglas production function is
a
a
y = x1 1 x a 2  xnn .
2

(kx1 ) a1 (kx 2 ) a 2 ...
Returns-to-Scale
 Q:

Can a technology exhibit
increasing returns-to-scale even
though all of its marginal products
are d...
Returns-to-Scale
 Q:

Can a technology exhibit
increasing returns-to-scale even if all
of its marginal products are
dimin...
Returns-to-Scale
a1 a 2
2/ 3 2/ 3
y = x1 x 2 = x1 x 2

4
a1 + a 2 = > 1 so this technology exhibits
3

increasing returns-...
Returns-to-Scale
a1 a 2
2/ 3 2/ 3
y = x1 x 2 = x1 x 2

4
a1 + a 2 = > 1 so this technology exhibits
3

increasing returns-...
Returns-to-Scale
a1 a 2
2/ 3 2/ 3
y = x1 x 2 = x1 x 2

4
a1 + a 2 = > 1 so this technology exhibits
3

increasing returns-...
Returns-to-Scale
 So

a technology can exhibit
increasing returns-to-scale even if all
of its marginal products are
dimin...
Returns-to-Scale
A

marginal product is the rate-ofchange of output as one input level
increases, holding all other input...
Returns-to-Scale
 When

all input levels are increased
proportionately, there need be no
diminution of marginal products
...
Technical Rate-of-Substitution
 At

what rate can a firm substitute
one input for another without
changing its output lev...
Technical Rate-of-Substitution
x2

x'2

y≡100
x'
1

x1
Technical Rate-of-Substitution
The slope is the rate at which
input 2 must be given up as
input 1’s level is increased so ...
Technical Rate-of-Substitution
 How

is a technical rate-of-substitution
computed?
Technical Rate-of-Substitution
 How

is a technical rate-of-substitution
computed?
 The production function is y = f ( x...
Technical Rate-of-Substitution
∂y
∂y
dy =
dx1 +
dx 2 .
∂ x1
∂ x2
But dy = 0 since there is to be no change
to the output l...
Technical Rate-of-Substitution
∂y
∂y
0=
dx1 +
dx 2
∂ x1
∂ x2
rearranges to
∂y
∂y
dx 2 = −
dx1
∂ x2
∂ x1
so

dx 2
∂ y / ∂ x...
Technical Rate-of-Substitution
dx 2
∂ y / ∂ x1
=−
dx1
∂ y / ∂ x2
is the rate at which input 2 must be given
up as input 1 ...
Technical Rate-of-Substitution; A
Cobb-Douglas Example
a b
y = f ( x1 , x 2 ) = x1 x 2

∂y
so ∂ y
a b −1
a −1 b
= bx1 x 2 ...
Technical Rate-of-Substitution; A
Cobb-Douglas Example
x2

1
2
a = and b =
3
3
( 1 / 3) x 2
x2
=−
=−
( 2 / 3 ) x1
2x1

1/ ...
Technical Rate-of-Substitution; A
Cobb-Douglas Example
x2

1
2
a = and b =
3
3
ax 2
( 1 / 3) x 2
x2
TRS = −
=−
=−
bx1
( 2 ...
Technical Rate-of-Substitution; A
Cobb-Douglas Example
x2

1
2
a = and b =
3
3
( 1 / 3) x 2
x2
=−
=−
( 2 / 3 ) x1
2x1

1/ ...
Well-Behaved Technologies
A

well-behaved technology is
monotonic, and
convex.
Well-Behaved Technologies Monotonicity
 Monotonicity:

More of any input
generates more output.

y

monotonic

x

y
not
m...
Well-Behaved Technologies Convexity
 Convexity:

If the input bundles x’
and x” both provide y units of output
then the m...
Well-Behaved Technologies Convexity
x2
'
x2

x"
2

y≡100
x'
1

x"
1

x1
Well-Behaved Technologies Convexity
x2
'
x2

(

tx' + (1 − t )x" , tx'2 + (1 − t )x"
1
1
2

x"
2

y≡100
x'
1

x"
1

x1

)
Well-Behaved Technologies Convexity
x2
'
x2

(

tx' + (1 − t )x" , tx'2 + (1 − t )x"
1
1
2

y≡120
y≡100

x"
2
x'
1

x"
1

...
Well-Behaved Technologies Convexity
Convexity implies that the TRS
increases (becomes less
negative) as x1 increases.

x2
...
Well-Behaved Technologies
higher output

x2

y≡200
y≡50 y≡100
x1
The Long-Run and the Short-Runs
 The

long-run is the circumstance in
which a firm is unrestricted in its
choice of all i...
The Long-Run and the Short-Runs
 Examples

of restrictions that place a
firm into a short-run:
temporarily being unable ...
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...
The Long-Run and the Short-Runs
 What

do short-run restrictions imply
for a firm’s technology?
 Suppose the short-run r...
The Long-Run and the Short-Runs
x2

y

x1
The Long-Run and the Short-Runs
x2

y

x1
The Long-Run and the Short-Runs

x2

y

x1
The Long-Run and the Short-Runs

x2

y
x1
The Long-Run and the Short-Runs

x2

y
x1
The Long-Run and the Short-Runs

x2
y
x1
The Long-Run and the Short-Runs

x2
y

x1
The Long-Run and the Short-Runs

x2y

x1
The Long-Run and the Short-Runs

y
x2
x1
The Long-Run and the Short-Runs

y

x2

x1
The Long-Run and the Short-Runs

y

x1
The Long-Run and the Short-Runs

y

x1
The Long-Run and the Short-Runs

y

x1

Four short-run production functions.
The Long-Run and the Short-Runs
1/3 1/ 3
y = x1 x 2 is the long-run production

function (both x1 and x2 are variable).
Th...
The Long-Run and the Short-Runs
y = x1 / 3 101 / 3
1
y = x1 / 3 51 / 3
1
y = x1 / 3 21 / 3
1

y

y = x1 / 3 11 / 3
1

x1

...
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Ch18

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Ch18

  1. 1. Chapter Eighteen Technology
  2. 2. 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.
  3. 3. 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?
  4. 4. 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, 9⋅3).
  5. 5. 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 1 ,  , xn )
  6. 6. Production Functions One input, one output Output Level y = f(x) is the production function. y’ y’ = f(x’) is the maximal output level obtainable from x’ input units. x’ Input Level x
  7. 7. Technology Sets A production plan is an input bundle and an output level; (x1, … , xn, y). A production plan is feasible if  The y ≤ f ( x1 ,  , xn ) collection of all feasible production plans is the technology set.
  8. 8. Technology Sets One input, one output Output Level y’ y” y = f(x) is the production function. y’ = f(x’) is the maximal output level obtainable from x’ input units. y” = f(x’) is an output level that is feasible from x’ input units. x’ x Input Level
  9. 9. Technology Sets The technology set is T = {( x1 ,, xn , y) | y ≤ f ( x1 ,, xn ) and x1 ≥ 0,, xn ≥ 0}.
  10. 10. Technology Sets One input, one output Output Level y’ The technology set y” x’ Input Level x
  11. 11. Technology Sets One input, one output Output Level Technically efficient plans y’ y” The technology Technically set inefficient plans x’ Input Level x
  12. 12. 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 1/3 1/3 y = f ( x1 , x 2 ) = 2x1 x 2 .
  13. 13. Technologies with Multiple Inputs  E.g. the maximal output level possible from the input bundle (x1, x2) = (1, 8) is y = 2x1/ 3x1/ 3 = 2 × 11/ 3× 81/ 3 = 2 × 1 × 2 = 4. 1 2  And the maximal output level possible from (x1,x2) = (8,8) is 1/ 3 1/ 3 1/ 3 1/ 3 y = 2x1 x 2 = 2 × 8 × 8 = 2 × 2 × 2 = 8.
  14. 14. Technologies with Multiple Inputs Output, y x2 (8,1) x1 (8,8)
  15. 15. 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.
  16. 16. Isoquants with Two Variable Inputs x2 y≡ 8 y≡ 4 x1
  17. 17. 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.
  18. 18. Isoquants with Two Variable Inputs Output, y y≡ 8 x2 y ≡ 4 x1
  19. 19. Isoquants with Two Variable Inputs  More isoquants tell us more about the technology.
  20. 20. Isoquants with Two Variable Inputs x2 y≡ 8 y≡ 6 y≡ 4 y≡ 2 x1
  21. 21. Isoquants with Two Variable Inputs Output, y y≡ 8 y≡ 6 x2 y ≡ 4 y≡ 2 x1
  22. 22. 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. 1 1  E.g. y = f ( x1 , x2 ) = 2 x1 / 3 x2 / 3
  23. 23. Technologies with Multiple Inputs x2 y x1
  24. 24. Technologies with Multiple Inputs x2 y x1
  25. 25. Technologies with Multiple Inputs x2 y x1
  26. 26. Technologies with Multiple Inputs x2 y x1
  27. 27. Technologies with Multiple Inputs x2 y x1
  28. 28. Technologies with Multiple Inputs x2 y x1
  29. 29. Technologies with Multiple Inputs y x1
  30. 30. Technologies with Multiple Inputs y x1
  31. 31. Technologies with Multiple Inputs y x1
  32. 32. Technologies with Multiple Inputs y x1
  33. 33. Technologies with Multiple Inputs y x1
  34. 34. Technologies with Multiple Inputs y x1
  35. 35. Technologies with Multiple Inputs y x1
  36. 36. Technologies with Multiple Inputs y x1
  37. 37. Technologies with Multiple Inputs y x1
  38. 38. Technologies with Multiple Inputs y x1
  39. 39. Cobb-Douglas Technologies A Cobb-Douglas production function is of the form a1 a 2 an y = A x1 x 2 ×× xn .  E.g. with 1/ 3 1/ 3 y = x1 x 2 1 1 n = 2, A = 1, a1 = and a 2 = . 3 3
  40. 40. Cobb-Douglas Technologies x2 All isoquants are hyperbolic, asymptoting to, but never touching any axis. a1 a 2 y = x1 x 2 x1
  41. 41. Cobb-Douglas Technologies x2 All isoquants are hyperbolic, asymptoting to, but never touching any axis. a1 a 2 y = x1 x 2 a1 a 2 x1 x 2 = y" x1
  42. 42. Cobb-Douglas Technologies x2 All isoquants are hyperbolic, asymptoting to, but never touching any axis. a1 a 2 y = x1 x 2 a1 a 2 x1 x 2 = y" a1 a 2 x1 x 2 = y' x1
  43. 43. Cobb-Douglas Technologies x2 All isoquants are hyperbolic, asymptoting to, but never touching any axis. y" > y' a1 a 2 y = x1 x 2 a1 a 2 x1 x 2 = y" a1 a 2 x1 x 2 = y' x1
  44. 44. Fixed-Proportions Technologies A fixed-proportions production function is of the form y = min{ a1 x1 , a 2x 2 ,, an xn }.  E.g. with y = min{ x1 , 2x 2 } n = 2, a1 = 1 and a 2 = 2.
  45. 45. Fixed-Proportions Technologies y = min{ x1 , 2x 2 } x2 x1 = 2x2 7 4 2 4 8 min{x1,2x2} = 14 min{x1,2x2} = 8 min{x1,2x2} = 4 14 x1
  46. 46. Perfect-Substitutes Technologies A perfect-substitutes production function is of the form y = a1 x1 + a 2x 2 +  + an xn .  E.g. with y = x1 + 3x 2 n = 2, a1 = 1 and a 2 = 3.
  47. 47. Perfect-Substitution Technologies y = x1 + 3x 2 x2 x1 + 3x2 = 18 x1 + 3x2 = 36 x1 + 3x2 = 48 8 6 3 All are linear and parallel 9 18 24 x1
  48. 48. Marginal (Physical) Products y = f ( x 1 ,  , xn )  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 MP i = ∂ xi
  49. 49. Marginal (Physical) Products E.g. if 1/ 3 2 / 3 y = f ( x1 , x 2 ) = x1 x 2 then the marginal product of input 1 is
  50. 50. Marginal (Physical) Products E.g. if 1/ 3 2 / 3 y = f ( x1 , x 2 ) = x1 x 2 then the marginal product of input 1 is ∂ y 1 − 2/ 3 2/ 3 MP1 = = x1 x 2 ∂ x1 3
  51. 51. Marginal (Physical) Products E.g. if 1/ 3 2 / 3 y = f ( x1 , x 2 ) = x1 x 2 then the marginal product of input 1 is ∂ y 1 − 2/ 3 2/ 3 MP1 = = x1 x 2 ∂ x1 3 and the marginal product of input 2 is
  52. 52. Marginal (Physical) Products E.g. if 1/ 3 2 / 3 y = f ( x1 , x 2 ) = x1 x 2 then the marginal product of input 1 is ∂ y 1 − 2/ 3 2/ 3 MP1 = = x1 x 2 ∂ x1 3 and the marginal product of input 2 is ∂ y 2 1/ 3 −1/ 3 MP2 = = x1 x 2 . ∂ x2 3
  53. 53. Marginal (Physical) Products Typically the marginal product of one input depends upon the amount used of other inputs. E.g. if 1 − 2/ 3 2/ 3 MP1 = x1 x 2 then, 3 1 − 2/ 3 2/ 3 4 − 2/ 3 8 = x1 if x2 = 8, MP1 = x1 3 3 and if x2 = 27 then 1 − 2/ 3 2/ 3 − MP1 = x1 27 = 3x1 2 / 3 . 3
  54. 54. 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 ∂ MP i ∂  ∂ y  ∂ 2y  =  ∂ x  = ∂ x 2 < 0. ∂ xi ∂ xi  i   i
  55. 55. Marginal (Physical) Products 1/ 3 2 / 3 E.g. if y = x1 x 2 then 1 − 2/ 3 2/ 3 2 1/ 3 −1/ 3 MP1 = x1 x 2 and MP2 = x1 x 2 3 3
  56. 56. Marginal (Physical) Products 1/ 3 2 / 3 E.g. if y = x1 x 2 then 1 − 2/ 3 2/ 3 2 1/ 3 −1/ 3 MP1 = x1 x 2 and MP2 = x1 x 2 so 3 3 ∂ MP1 2 − 5 / 3 2/ 3 = − x1 x 2 < 0 ∂ x1 9
  57. 57. Marginal (Physical) Products 1/ 3 2 / 3 E.g. if y = x1 x 2 then 1 − 2/ 3 2/ 3 2 1/ 3 −1/ 3 MP1 = x1 x 2 and MP2 = x1 x 2 so 3 3 ∂ MP1 2 − 5 / 3 2/ 3 = − x1 x 2 < 0 ∂ x1 9 and ∂ MP 2 = − 2 x1/ 3x − 4 / 3 < 0. 1 2 ∂ x2 9
  58. 58. Marginal (Physical) Products 1/ 3 2 / 3 E.g. if y = x1 x 2 then 1 − 2/ 3 2/ 3 2 1/ 3 −1/ 3 MP1 = x1 x 2 and MP2 = x1 x 2 so 3 3 ∂ MP1 2 − 5 / 3 2/ 3 = − x1 x 2 < 0 ∂ x1 9 and ∂ MP 2 = − 2 x1/ 3x − 4 / 3 < 0. 1 2 ∂ x2 9 Both marginal products are diminishing.
  59. 59. 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).
  60. 60. Returns-to-Scale If, for any input bundle (x1,…,xn), f (kx1 , kx 2 ,, kxn ) = kf ( x1 , x 2 ,, xn ) then the technology described by the production function f exhibits constant returns-to-scale. E.g. (k = 2) doubling all input levels doubles the output level.
  61. 61. Returns-to-Scale One input, one output Output Level y = f(x) 2y’ Constant returns-to-scale y’ x’ 2x’ Input Level x
  62. 62. Returns-to-Scale If, for any input bundle (x1,…,xn), f (kx1 , kx 2 ,, kxn ) < kf ( x1 , x 2 ,, xn ) then the technology exhibits diminishing returns-to-scale. E.g. (k = 2) doubling all input levels less than doubles the output level.
  63. 63. Returns-to-Scale One input, one output Output Level 2f(x’) y = f(x) f(2x’) Decreasing returns-to-scale f(x’) x’ 2x’ Input Level x
  64. 64. Returns-to-Scale If, for any input bundle (x1,…,xn), f (kx1 , kx 2 ,, kxn ) > kf ( x1 , x 2 ,, xn ) then the technology exhibits increasing returns-to-scale. E.g. (k = 2) doubling all input levels more than doubles the output level.
  65. 65. Returns-to-Scale One input, one output Output Level Increasing returns-to-scale y = f(x) f(2x’) 2f(x’) f(x’) x’ 2x’ Input Level x
  66. 66. Returns-to-Scale A single technology can ‘locally’ exhibit different returns-to-scale.
  67. 67. Returns-to-Scale One input, one output Output Level y = f(x) Increasing returns-to-scale Decreasing returns-to-scale x Input Level
  68. 68. Examples of Returns-to-Scale The perfect-substitutes production function is y = a1 x1 + a 2x 2 +  + an xn . Expand all input levels proportionately by k. The output level becomes a1 (kx1 ) + a 2 (kx 2 ) +  + an (kxn )
  69. 69. Examples of Returns-to-Scale The perfect-substitutes production function is y = a1 x1 + a 2x 2 +  + an xn . Expand all input levels proportionately by k. The output level becomes a1 (kx1 ) + a 2 (kx 2 ) +  + an (kxn ) = k( a1x1 + a 2x 2 +  + anxn )
  70. 70. Examples of Returns-to-Scale The perfect-substitutes production function is y = a1 x1 + a 2x 2 +  + an xn . Expand all input levels proportionately by k. The output level becomes a1 (kx1 ) + a 2 (kx 2 ) +  + an (kxn ) = k( a1x1 + a 2x 2 +  + anxn ) = ky. The perfect-substitutes production function exhibits constant returns-to-scale.
  71. 71. Examples of Returns-to-Scale The perfect-complements production function is y = min{ a1 x1 , a 2x 2 ,  , an xn }. Expand all input levels proportionately by k. The output level becomes min{ a1 (kx1 ), a 2 (kx 2 ),  , an (kxn )}
  72. 72. Examples of Returns-to-Scale The perfect-complements production function is y = min{ a1 x1 , a 2x 2 ,  , an xn }. Expand all input levels proportionately by k. The output level becomes min{ a1 (kx1 ), a 2 (kx 2 ),  , an (kxn )} = k(min{ a1x1 , a 2x 2 ,  , anxn })
  73. 73. Examples of Returns-to-Scale The perfect-complements production function is y = min{ a1 x1 , a 2x 2 ,  , an xn }. Expand all input levels proportionately by k. The output level becomes min{ a1 (kx1 ), a 2 (kx 2 ),  , an (kxn )} = k(min{ a1x1 , a 2x 2 ,  , anxn }) = ky. The perfect-complements production function exhibits constant returns-to-scale.
  74. 74. Examples of Returns-to-Scale The Cobb-Douglas production function is a a y = x1 1 x a 2  xnn . 2 Expand all input levels proportionately by k. The output level becomes (kx1 ) a1 (kx 2 ) a2 (kxn ) an
  75. 75. Examples of Returns-to-Scale The Cobb-Douglas production function is a a y = x1 1 x a 2  xnn . 2 Expand all input levels proportionately by k. The output level becomes (kx1 ) a1 (kx 2 ) a2 (kxn ) an a1 a 2 an a1 a 2 an = k k k x x x
  76. 76. Examples of Returns-to-Scale The Cobb-Douglas production function is a a y = x1 1 x a 2  xnn . 2 Expand all input levels proportionately by k. The output level becomes (kx1 ) a1 (kx 2 ) a 2 (kxn ) an = k a1k a 2 k an x a1 x a 2 x an a a = k a1 +a 2 ++an x1 1 x a 2 xnn 2
  77. 77. Examples of Returns-to-Scale The Cobb-Douglas production function is a a y = x1 1 x a 2  xnn . 2 Expand all input levels proportionately by k. The output level becomes (kx1 ) a1 (kx 2 ) a 2 (kxn ) an = k a1k a 2 k an x a1 x a 2 x an a a = k a1 +a 2 ++an x1 1 x a 2 xnn 2 = k a1 ++an y.
  78. 78. Examples of Returns-to-Scale The Cobb-Douglas production function is a a y = x1 1 x a 2  xnn . 2 (kx1 ) a1 (kx 2 ) a 2 (kxn ) an = k a1 ++an y. The Cobb-Douglas technology’s returnsto-scale is constant if a1+ … + an = 1
  79. 79. Examples of Returns-to-Scale The Cobb-Douglas production function is a a y = x1 1 x a 2  xnn . 2 (kx1 ) a1 (kx 2 ) a 2 (kxn ) an = k a1 ++an y. The Cobb-Douglas technology’s returnsto-scale is constant if a1+ … + an = 1 increasing if a1+ … + an > 1
  80. 80. Examples of Returns-to-Scale The Cobb-Douglas production function is a a y = x1 1 x a 2  xnn . 2 (kx1 ) a1 (kx 2 ) a 2 (kxn ) an = k a1 ++an y. The Cobb-Douglas technology’s returnsto-scale is constant if a1+ … + an = 1 increasing if a1+ … + an > 1 decreasing if a1+ … + an < 1.
  81. 81. Returns-to-Scale  Q: Can a technology exhibit increasing returns-to-scale even though all of its marginal products are diminishing?
  82. 82. Returns-to-Scale  Q: Can a technology exhibit increasing returns-to-scale even if all of its marginal products are diminishing?  A: Yes. 2  E.g. y = x1 / 3x 2 / 3 . 2
  83. 83. Returns-to-Scale a1 a 2 2/ 3 2/ 3 y = x1 x 2 = x1 x 2 4 a1 + a 2 = > 1 so this technology exhibits 3 increasing returns-to-scale.
  84. 84. Returns-to-Scale a1 a 2 2/ 3 2/ 3 y = x1 x 2 = x1 x 2 4 a1 + a 2 = > 1 so this technology exhibits 3 increasing returns-to-scale. 2 −1/ 3 2/ 3 But MP1 = x1 x 2 diminishes as x1 3 increases
  85. 85. Returns-to-Scale a1 a 2 2/ 3 2/ 3 y = x1 x 2 = x1 x 2 4 a1 + a 2 = > 1 so this technology exhibits 3 increasing returns-to-scale. 2 −1/ 3 2/ 3 But MP1 = x1 x 2 diminishes as x1 3 increases and 2 2/ 3 −1/ 3 MP2 = x1 x 2 diminishes as x1 3 increases.
  86. 86. Returns-to-Scale  So a technology can exhibit increasing returns-to-scale even if all of its marginal products are diminishing. Why?
  87. 87. Returns-to-Scale A marginal product is the rate-ofchange 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.
  88. 88. 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.
  89. 89. Technical Rate-of-Substitution  At what rate can a firm substitute one input for another without changing its output level?
  90. 90. Technical Rate-of-Substitution x2 x'2 y≡100 x' 1 x1
  91. 91. Technical Rate-of-Substitution 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 x'2 y≡100 x' 1 x1
  92. 92. Technical Rate-of-Substitution  How is a technical rate-of-substitution computed?
  93. 93. Technical Rate-of-Substitution  How is a technical rate-of-substitution computed?  The production function is y = f ( x1 , x 2 ).  A small change (dx1, dx2) in the input bundle causes a change to the output level of ∂y ∂y dy = dx1 + dx 2 . ∂ x1 ∂ x2
  94. 94. Technical Rate-of-Substitution ∂y ∂y dy = dx1 + dx 2 . ∂ x1 ∂ x2 But dy = 0 since there is to be no change to the output level, so the changes dx 1 and dx2 to the input levels must satisfy ∂y ∂y 0= dx1 + dx 2 . ∂ x1 ∂ x2
  95. 95. Technical Rate-of-Substitution ∂y ∂y 0= dx1 + dx 2 ∂ x1 ∂ x2 rearranges to ∂y ∂y dx 2 = − dx1 ∂ x2 ∂ x1 so dx 2 ∂ y / ∂ x1 =− . dx1 ∂ y / ∂ x2
  96. 96. Technical Rate-of-Substitution dx 2 ∂ y / ∂ x1 =− dx1 ∂ y / ∂ x2 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.
  97. 97. Technical Rate-of-Substitution; A Cobb-Douglas Example a b y = f ( x1 , x 2 ) = x1 x 2 ∂y so ∂ y a b −1 a −1 b = bx1 x 2 . = ax1 x 2 and ∂ x2 ∂ x1 The technical rate-of-substitution is a −1 b dx 2 ∂ y / ∂ x1 ax1 x 2 ax 2 =− =− =− . a dx1 ∂ y / ∂ x2 bx1 bx1 xb −1 2
  98. 98. Technical Rate-of-Substitution; A Cobb-Douglas Example x2 1 2 a = and b = 3 3 ( 1 / 3) x 2 x2 =− =− ( 2 / 3 ) x1 2x1 1/ 3 2 / 3 y = x1 x 2 ; ax 2 TRS = − bx1 x1
  99. 99. Technical Rate-of-Substitution; A Cobb-Douglas Example x2 1 2 a = and b = 3 3 ax 2 ( 1 / 3) x 2 x2 TRS = − =− =− bx1 ( 2 / 3 ) x1 2x1 x2 8 TRS = − =− = −1 2x1 2×4 1/ 3 2 / 3 y = x1 x 2 ; 8 4 x1
  100. 100. Technical Rate-of-Substitution; A Cobb-Douglas Example x2 1 2 a = and b = 3 3 ( 1 / 3) x 2 x2 =− =− ( 2 / 3 ) x1 2x1 1/ 3 2 / 3 y = x1 x 2 ; ax 2 TRS = − bx1 x2 6 1 TRS = − =− =− 2x1 2 × 12 4 6 12 x1
  101. 101. Well-Behaved Technologies A well-behaved technology is monotonic, and convex.
  102. 102. Well-Behaved Technologies Monotonicity  Monotonicity: More of any input generates more output. y monotonic x y not monotonic x
  103. 103. 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.
  104. 104. Well-Behaved Technologies Convexity x2 ' x2 x" 2 y≡100 x' 1 x" 1 x1
  105. 105. Well-Behaved Technologies Convexity x2 ' x2 ( tx' + (1 − t )x" , tx'2 + (1 − t )x" 1 1 2 x" 2 y≡100 x' 1 x" 1 x1 )
  106. 106. Well-Behaved Technologies Convexity x2 ' x2 ( tx' + (1 − t )x" , tx'2 + (1 − t )x" 1 1 2 y≡120 y≡100 x" 2 x' 1 x" 1 x1 )
  107. 107. Well-Behaved Technologies Convexity Convexity implies that the TRS increases (becomes less negative) as x1 increases. x2 ' x2 x" 2 x' 1 x" 1 x1
  108. 108. Well-Behaved Technologies higher output x2 y≡200 y≡50 y≡100 x1
  109. 109. 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.
  110. 110. 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.
  111. 111. 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.
  112. 112. 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.
  113. 113. The Long-Run and the Short-Runs x2 y x1
  114. 114. The Long-Run and the Short-Runs x2 y x1
  115. 115. The Long-Run and the Short-Runs x2 y x1
  116. 116. The Long-Run and the Short-Runs x2 y x1
  117. 117. The Long-Run and the Short-Runs x2 y x1
  118. 118. The Long-Run and the Short-Runs x2 y x1
  119. 119. The Long-Run and the Short-Runs x2 y x1
  120. 120. The Long-Run and the Short-Runs x2y x1
  121. 121. The Long-Run and the Short-Runs y x2 x1
  122. 122. The Long-Run and the Short-Runs y x2 x1
  123. 123. The Long-Run and the Short-Runs y x1
  124. 124. The Long-Run and the Short-Runs y x1
  125. 125. The Long-Run and the Short-Runs y x1 Four short-run production functions.
  126. 126. The Long-Run and the Short-Runs 1/3 1/ 3 y = x1 x 2 is the long-run production function (both x1 and x2 are variable). The short-run production function when x2 ≡ 1 is y = x1 / 3 11 / 3 = x1 / 3 . 1 1 The short-run production function when x2 ≡ 10 is y = x1 / 3 101 / 3 = 2 ⋅ 15x1 / 3 . 1 1
  127. 127. The Long-Run and the Short-Runs y = x1 / 3 101 / 3 1 y = x1 / 3 51 / 3 1 y = x1 / 3 21 / 3 1 y y = x1 / 3 11 / 3 1 x1 Four short-run production functions.

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