Technology
Molly W. Dahl
Georgetown University
Econ 101 – Spring 2009
1
Technologies

A technology is a process by which inputs
are converted to an output.
 E.g.
labor, a computer, a projector, electricity,
software, chalk, a blackboard are all being
used to produce this lecture.
2
Production Functions
xi denotes the amount of input i
 y denotes the output level.
 The technology’s production function
states the maximum amount of output
possible from an input bundle.

y  f ( x1 ,, xn )
3
Production Functions
One input, one output
Output Level
y’
y = f(x) is the
production
function.
y’ = f(x’) is the maximum
output obtainable from x’
input units.
x’
Input Level
x
4
Technology Sets
One input, one output
Output Level
y’
y”
y = f(x) is the
production
function.
y’ = f(x’) is the maximum
output obtainable from x’
input units.
y” = f(x’) is an output level
that is feasible from x’
input units.
x’
x
Input Level
5
Technology Sets
One input, one output
Output Level
y’
The technology
set
y”
x’
Input Level
x
6
Technology Sets
One input, one output
Output Level
Technically
efficient plans
y’
y”
The technology
Technically set
inefficient
plans
x’
Input Level
x
7
Technologies with Multiple Inputs
What does a technology look like when
there is more than one input?
 Suppose the production function is

1/3 1/3
y  f ( x1 , x 2 )  2x1 x 2 .
8
Technologies with Multiple Inputs
The isoquant is the set of all input bundles
that yield the same maximum output level
y.
 More isoquants tell us more about the
technology.

9
Isoquants with Two Variable Inputs
x2
x2
y 
y 
y 
y 
x1
x1
10
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.

y
1/ 3 1/ 3
f ( x1 , x2 )  2 x1 x2
11
Technologies with Multiple Inputs
x2
x2
y
x1
x1
12
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
13
Cobb-Douglas Technologies
x2
All isoquants are hyperbolic,
asymptoting to, but never
touching any axis.
y" > y'
a1 a 2
y  x1 x 2
x1
14
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.
15
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
16
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.
17
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
18
Well-Behaved Technologies

A well-behaved technology is
 monotonic,
and
 convex.
19
Well-Behaved Technologies Monotonicity

Monotonicity: More of any input generates
more output.
y
y
monotonic
not
monotonic
x
x
20
Well-Behaved Technologies Monotonicity
higher output
x2
y
y y
x1
21
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.
22
Well-Behaved Technologies Convexity
x2
x'2

tx'1  (1  t )x"1 , tx'2  (1  t )x"2
x"2

y
x'1
x"1
x1
23
Well-Behaved Technologies Convexity
x2
x'2

tx'1  (1  t )x"1 , tx'2  (1  t )x"2

y
y
x"2
x'1
x"1
x1
24
Marginal (Physical) Products
y  f ( x1 ,, 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
MPi 
 xi
25
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
26
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
27
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
28
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
 MPi
   y   2y

 

 0.
2
 xi
 xi   xi   xi
29
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
30
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
31
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
and
3
3
 MP1
2  5 / 3 2/ 3
  x1 x 2  0
 x1
9
 MP2
2 1/ 3  4 / 3
  x1 x 2
 0.
 x2
9
Both marginal products are diminishing.
32
Technical Rate-of-Substitution

At what rate can a firm substitute one input
for another without changing its output
level?
33
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
x'1
x1
34
Technical Rate-of-Substitution

How is a technical rate-of-substitution
computed?
35
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
36
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 dx1
and dx2 to the input levels must satisfy
y
y
0
dx1 
dx 2 .
 x1
 x2
37
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
38
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.
39
TRS: A Cobb-Douglas Example
a b
y  f ( x1 , x 2 )  x1 x 2
so  y
 x1
a1 b
 ax1 x 2 and
y
a b 1
 bx1 x 2 .
 x2
The technical rate-of-substitution is
a1 b
dx 2
 y /  x1
ax1 x 2
ax 2



.
1
dx1
 y /  x2
bx1
bx1axb
2
40
The Long-Run and the ShortRun
In the long-run a firm is unrestricted in its
choice of all input levels.
 There are many possible short-runs.
 In the short-run a firm is restricted in some
way in its choice of at least one input level.

41
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
equal proportion

 e.g.
all input levels doubled, or halved
42
Constant Returns-to-Scale
If, for any input bundle (x1,…,xn),
f (kx1 , kx 2 ,, kxn )  kf ( x1 , x 2 ,, xn )
then the technology exhibits constant
returns-to-scale (CRS).
E.g. (k = 2) If doubling all input levels
doubles the output level, the technology
exhibits CRS.
43
Constant Returns-to-Scale
One input, one output
Output Level
y = f(x)
2y’
Constant
returns-to-scale
y’
x’
2x’
Input Level
x
44
Decreasing Returns-to-Scale
If, for any input bundle (x1,…,xn),
f (kx1 , kx 2 ,, kxn )  kf ( x1 , x 2 ,, xn )
then the technology exhibits decreasing
returns-to-scale (DRS).
E.g. (k = 2) If doubling all input levels
less than doubles the output level, the
technology exhibits DRS.
45
Decreasing 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
46
Increasing 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 (IRS).
E.g. (k = 2) If doubling all input levels
more than doubles the output level, the
technology exhibits IRS.
47
Increasing 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
48
Examples of Returns-to-Scale
The Cobb-Douglas production function is
2  x an .
y  x1a1 xa
n
2
Expand all input levels proportionately
by k. The output level becomes
(kx1 )
a1
(kx 2 )
a2
(kxn )
an
49
Examples of Returns-to-Scale
The Cobb-Douglas production function is
2  x an .
y  x1a1 xa
n
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
50
Examples of Returns-to-Scale
The Cobb-Douglas production function is
2  x an .
y  x1a1 xa
n
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
2 x an
 k a1  a 2  an x1a1 x a
n
2
51
Examples of Returns-to-Scale
The Cobb-Douglas production function is
2  x an .
y  x1a1 xa
n
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
2 x an
 k a1  a 2  an x1a1 x a
n
2
 k a1  an y.
52
Examples of Returns-to-Scale
The Cobb-Douglas production function is
2  x an .
y  x1a1 xa
n
2
(kx1 )a1 (kx 2 )a 2 (kxn )an  ka1  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.
53
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 )
54
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 )
55
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.
56
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 )}
57
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 })
58
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.
59