The theory of the firm:
Technology
Lectures in Microeconomic Theory
Fall 2010, Part 2
07.07.2010
G.B. Asheim, ECON4230-35, #2
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Specification of technology

A firm produces outputs from inputs. We need to
specify what combinations of inputs and outputs
that are technologically feasible.
Good j :
y  y oj  y ij
j


Net
output
Output
Input
Production plan : y  ( y1 ,  , y j ,  , yn )

Production (possibilities) set: Y
Negative if
input
Positive if
output
y  ( y1 ,  , y j ,  , yn ) is feasible if and only if y  Y
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G.B. Asheim, ECON4230-35, #2
Proporties of production sets

Y is nonempty and closed

Monotonicity: if y  Y & y  y, then y  Y

“No free lunch”
Non-incr RTS
Convex
”Free lunch”


Y is convex
Non-increasing returns to scale
Non-incr RTS
Convex
y2
y2
Y
Y
y1
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G.B. Asheim, ECON4230-35, #2
y1
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Properties of production sets (cont.)

y2
Not non-incr RTS
Not convex
Y yields
possibility
for inaction
Y
Not non-incr RTS
Convex
No poss for inaction
y2 Not non-incr RTS
Not convex
Poss for inaction
Y
y1
y2
Y
y1
y1
Sunk cost
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G.B. Asheim, ECON4230-35, #2
Only one output i.e. : y  ( y,x) 

Input requirement set: V ( y )  {x | ( y, x)  Y }

Isoquant: Q ( y )  {x | x  V ( y ) & x  V ( y ) for y   y}
Analysis with only one output
Production function Illustrations with only two inputs

From input requirement set to production function

From production function to input requirement set
f (x)  max{ y | x  V ( y )}
V ( y )  {x | f ( x)  y}
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G.B. Asheim, ECON4230-35, #2
Input requirement set
x2
Monotonicity
Convexity
y  2 further out if
I
Input
vector that
h lleads
d to y  1.
1 decreasing RTS
y  2 if constant RTS
y  leads
2 further
in if
Input vector that also
to y  1.
isoquant
07.07.2010
y  1 increasing RTS
x1
G.B. Asheim, ECON4230-35, #2
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Convexity
0  t 1
Y convex : y  Y and y   Y imply ty  (1  t ) y   Y
V ( y ) convex : x  V ( y ) and x  V ( y ) imply
tx  (1  t )x  V ( y )
f quasi - concave :
f ( x)  f ( x) implies
p
f (tx  (1  t ) x)  f ( x)

A convex production set Y implies that the associated
input requirement set V(y) is convex.

A convex input requirement set V(y) is equivalent to a
quasi-concave prodution function f(x) .
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G.B. Asheim, ECON4230-35, #2
The technical rate of substitution

Slope of the isoquant

If x1 is increased with one unit, how much can
x2 be decreased while keeping output constant?
x2
y  f ( x1 , x2 )
0  dy 
f
x1
dx1  xf2 dx2
x1
TRS 
dx2
dx1
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
y  f ( x1 , x2 )
f
x1
f
x 2
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G.B. Asheim, ECON4230-35, #2
The elasticity of substitution

Curvature of the isoquant

How easy is it to substitute one input for another?
x2

x1
x2
x1
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
 ( x2 / x1 )
x2 / x1
TRS
TRS
d ln y  1y dyy
y  f ( x1 , x2 )
y
 
x
TRS d x12
x2
dTRS
x1
d ln( x2 / x1 )

d ln TRS
d ln x  1x dx
y  f ( x1 , x2 )

d ln y dy x

d ln x dx y
y  f ( x1 , x2 )
G.B. Asheim, ECON4230-35, #2
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3

Returns to scale (RTS)
A technology
exhibits constant RTS if any of the following holds:
y
y  Y  ty  Y , for all t  0
x  V ( y )  tx  V (ty ), for all t  0
f (tx)  tf ( x), for all t  0
x
Y
The production function is homogeneous of degree 1.

A technology exhibits increasing RTS if:
f (tx)  tf ( x), for all t  1

A technology exhibits decreasing RTS if:
f (tx)  tf ( x), for all t  1
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G.B. Asheim, ECON4230-35, #2
Prod. fn.s with constant RTS ― Examples

Cobb-Douglas prod. fn.
f ( x1 , x2 )  x1a  x2b , 0  a, b  1, a  b  1
What is the TRS?
TRS  
f
x1
f
x2

a x2
b x1
 
 b 
f
x1
 ax1a 1  x2b  a
x2 b
x1
f
x2
 bx1a  x2b 1
x1 a
x2
What is the elasticity of substitution?

x2
x1
d ln( x2 / x1 )
1
d ln TRS
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  ba TRS
ln xx12  ln ba  ln TRS
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G.B. Asheim, ECON4230-35, #2
Prod. fn.s with constant RTS ― Examples

Constant-elasticty-of-substitution (CES) prod. fn.

 1


 1  1
f ( x1 , x2 )  ax1   bx2
,   0,   1
What is the TRS?
TRS  
ff
x1
f
x 2
1

ax 
   2 
b  x1 
1
1
1
1
f
x1
 ay  1 x1 
f
x 2
 by  1 x2 
What is the elasticity of substitution?
d ln( x2 / x1 )

d ln TRS
07.07.2010
x2
x1
  ba TRS 

ln xx12   ln ba   ln TRS
G.B. Asheim, ECON4230-35, #2
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Prod. fn.s with constant RTS ― Examples

Leontieff prod. fn.
f ( x1 , x2 )  min ax1 , bx2 , a  0, b  0
What is the TRS?
x2
Not defined
What is the elasticity
of substitution?
x1
0
07.07.2010
G.B. Asheim, ECON4230-35, #2
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Elasticity of scale

Curvature of production fn. when scale is changed
e( x) 
df ( tx )
f ( tx )
dt
t

t 1
df (tx) t
dt f (tx) t 1
Constant RTS at x : e( x)  1
Increasing RTS at x : e(x)  1
Decreasing RTS at x : e(x)  1
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G.B. Asheim, ECON4230-35, #2
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Homogeneous and homothetic technologies

f(x) is homogeneous of degree k if
f (tx)  t k f ( x), for all t  0


f(x) is homothetic if it can be written as f(x)  g(h(x)),
where h() is homogeneous of degree 1 and g() is a
monotone function ( z  z   g ( z )  g ( z ) ).
Examples:
f ( x1 , x2 )  x1a  x2b , 0  a, b  1, a  b  1

 1
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
k
 1  1
f ( x1 , x2 )  ax1   bx2
,   0,   1, k  1
G.B. Asheim, ECON4230-35, #2
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