Chapter 9
Production Functions
Chapter Overview
1. The Production Function
 Marginal and Average Products
 Isoquants
 The Marginal Rate of Technical Substitution
2. Returns to Scale
3. Technical Progress
4. Some Special Functional Forms
2
Key Concepts
Productive resources, such as labor and capital
equipment, that firms use to manufacture goods and
services are called inputs or factors of production.
The amount of goods and services produces by the firm
is the firm’s output.
Production transforms a set of inputs into a set of
outputs
Technology determines the quantity of output that is
3
feasible to attain for a given set of inputs.
Key Concepts
The production function tells us the maximum possible
output that can be attained by the firm for any given
quantity of inputs.
Production Function:
Q  f ( L, K )
• Q = output
• K = Capital
• L = Labor
4
The Production Function & Technical Efficiency
• Technically efficient: Sets of points in the
production function that maximizes output
given input (labor)
Q  f ( L, K )
• Technically inefficient: Sets of points that
produces less output than possible for a
given set of input (labor) Q  f ( L, K )
5
The Production Function & Technical Efficiency
6
Short run versus Long run
• Short run - a period of time so brief that at least
one factor of production cannot be varied
practically.
– Fixed input - a factor of production that
cannot be varied practically in the short run.
– Variable input - a factor of production whose
quantity can be changed readily by the firm
during the relevant time period.
• Long run - a lengthy enough period of time that
all inputs can be varied.
Short-Run Production
• In the short run, the firm’s production
function is
q = f(L, K)
– where q is output, L is workers, and K is the
fixed number of units of capital.
Total Product, Marginal Product, and Average Product
of Labor with Fixed Capital
Total Product of Labor
• Total product of labor- the amount of output
(or total product) that can be produced by a
given amount of labor.
Marginal Product of Labor
• Marginal product of labor (MPL ) - the change in
total output, Dq, resulting from using an extra unit
of labor, DL, holding other factors constant:
Dq
MPL 
DL
MPL = DQ/DL
• (holding constant all other inputs)
MPK = DQ/DK
• (holding constant all other inputs)
Marginal Physical Product
• Marginal physical product is the additional
output that can be produced by employing
one more unit of that input
– holding other inputs constant
q
marginal physical product of capital  MPk 
 fk
k
q
marginal physical product of labor  MPl 
 fl
l
Diminishing Marginal Productivity
• Marginal physical product depends on how
much of that input is used
• In general, we assume diminishing marginal
productivity
MPk  2 f
 2  f kk  0
k
k
MPl  2 f
 2  f ll  0
l
l
Average Product of Labor
• Average product of labor (APL ) - the ratio of
output, q, to the number of workers, L, used to
produce that output:
q
APL 
L
APL = Q/L
APK = Q/K
Production Relationships
with Variable Labor
Output, q, Units per day
(a)
C
110
90
B
56
0
4
6
11
L, Workers per day
(b)
APL, MPL
Diminishing Marginal
Returns sets in!
A
a
20
b
15
Average product, APL
Marginal product, MPL
c
0
4
6
11
L, Workers per day
Total Product
16
Law of Diminishing Marginal Returns
If a firm keeps increasing an input, holding all
other inputs and technology constant, the
corresponding increases in output will become
smaller eventually.
– That is, if only one input is increased, the marginal
product of that input will diminish eventually.
Example: Short run Production
Function
• Suppose the production function for a firm can
be represented by
q = f(k,l) = 600k 2l2 - k 3l3
• To construct MPl and APl, we must assume a
value for k
– let k = 10
• The production function becomes
q = 60,000l2 - 1000l3
Long-Run Production
• In the long run both labor and capital are
variable inputs.
• It is possible to substitute one input for the
other while holding output constant.
Isoquants
• Isoquant - a curve that shows the efficient
combinations of labor and capital that can
produce a single (iso) level of output
(quantity).
• Equation for an isoquant:
q = f (L, K).
Isoquant Map
• Each isoquant represents a different level of
output
– output rises as we move northeast
k per period
q = 30
q = 20
l per period
Properties of Isoquants
1. The farther an isoquant is from the origin,
the greater the level of output.
2. Isoquants do not cross.
3. Isoquants slope downward
Substituting Inputs
• Marginal rate of technical substitution (MRTS) the number of extra units of one input needed to
replace one unit of another input that enables a
firm to keep the amount of output it produces
constant.
change in capital DK
MRTS 

change in labor
DL
Slope of Isoquant!
K
, Units of capital
How the Marginal Rate of Technical Substitution Varies
Along an Isoquant
a
16
DK = –6
b
10
DL = 1
–3
c
1
–2 1
7
5
4
d
e
–1
q = 10
1
0
1
2
3
4
5
6
7
8
9
L, WorkErs
10
Production Function With Two
Variable Inputs:
Moving from short run to long run
K
6
5
4
3
2
1
Q = f(L, K)
10
12
12
10
7
3
1
24
28
28
23
18
8
2
31
36
36
33
28
12
3
36
40
40
36
30
14
4
Q
40
42
40
36
30
14
5
39
40
36
33
28
12
6
L
Production Function With Two Variable Inputs
Production With Two Variable Inputs
• Isoquants show combinations of two
inputs that can produce the same level
of output.
• Firms will only use combinations of
two inputs that are in the economic
region of production.
Production With Two
Variable Inputs
Isoquants
K
6
5
4
3
2
1
Q
10
12
12
10
7
3
1
24
28
28
23
18
8
2
31
36
36
33
28
12
3
36
40
40
36
30
14
4
40
42
40
36
30
14
5
39
40
36
33
28
12
6 L
Economic Region of Production
• The firm should not use certain combinations of
outputs in the long run no matter how cheap they
are.
• These input combinations are represented by the
portion of the isoquant curve that has a positive
slope.
• A positively sloped isoquant means that merely to
maintain the same level of production, the firm
must use more of both the inputs if it increases its
use of one of the inputs.
• Ridge Lines:
– The lines connecting the points where the marginal
product of an input is equal to zero (one line for
each input) in the isoquant map and forming the
boundary for the economic region of production.
Production With Two Variable Inputs
Economic Region of Production
The Economic Region of Production
The economic region of production is the range in an
isoquant diagram where both inputs have a positive
marginal product.
It lies inside the ridge lines.
A profit maximizing firm will never try to produce
using input combinations outside the ridge lines.
MARGINAL RATE OF TECHNICAL SUBSTITUTION
Isoquants are downward
sloping and convex to the
origin.
The slope of the isoquant
(dK / dL) defines the
degree of substitutability of
the factors of production.
The slope of the isoquant
decreases (in absolute
terms) as we move
downwards along the
isoquant, showing the
increasingdifficulty in
substituting in substituting
K for L.
Production Functions—Two Special
Cases
Two extreme cases of production functions show
the possible range of input substitution in the
production process:
1) the case of perfect substitutes and
2) the fixed proportions production
function
The fixed-proportions production function
describes situations in which methods of
production are limited.
Production With Two Variable Inputs
Perfect Substitutes
•When the isoquants are straight
lines, the MRTS is constant.
• The rate at which K and L can
be substituted for each other is
the same no matter what level of
inputs is being used.
Perfect Complements
• When the isoquants are Lshaped, only one combination of
labor and capital can be used to
produce a given output.
• Adding more labor alone does
not increase output, nor does
adding more capital alone.
Substitutability of Inputs
Elasticity of Substitution
• A measure of how easy is it for a firm to
substitute labor for capital.
• It is the percentage change in the capitallabor ratio for every one percent change in
the MRTSL,K along an isoquant.
36
Elasticity of Substitution
• The elasticity of substitution () measures the
proportionate change in K/L relative to the
proportionate change in the MRTSx,y along an
isoquant
%D( K / L)
d ( K / L) MRTSk , l



%DMRTSk , l dMRTSk , l
K/L
• The value of  will always be positive because
K/L and MRTS move in the same direction.
Elasticity of Substitution
• Both MRTS and K/L will change as we
move from point A to point B
 is the ratio of these
proportional changes
K per period
RTSA
A
(k/l)A
(k/l)B
RTSB
B
q = q0
 measures the
curvature of the
isoquant
L per period
Elasticity of Substitution
• If  is high, the MRTS will not change much
relative to K/L
– the isoquant will be relatively flat
• If  is low, the MRTS will change by a substantial
amount as K/L changes
– the isoquant will be sharply curved
• It is possible for  to change along an isoquant
or as the scale of production changes
Elasticity of Substitution
K
"The shape of the
isoquant indicates the
degree of substitutability
of the inputs…"
=0
=1
 = 5
=
0
L
40
Elasticity of Substitution
• If we define the elasticity of substitution
between two inputs to be the proportionate
change in the ratio of the two inputs to the
proportionate change in MRTS, we need to
hold output and the levels of other inputs
constant
Returns to Scale
• How much will output increase when ALL
inputs increase by a particular amount?
•suppose that all inputs are doubled, would
output double?
%D(quantity of output)
Returns to Scale 
%D(quantity of all inputs)
42
Returns to Scale
• If the production function is given by q = f(k,l)
and all inputs are multiplied by the same
positive constant (t >1), then
Effect on Output Returns to Scale
f(tk,tl) = tf(k,l)
Constant
f(tk,tl) < tf(k,l)
Decreasing
f(tk,tl) > tf(k,l)
Increasing
Returns to Scale
• How
much will output increase when ALL inputs increase
by a particular amount?
• RTS = [%DQ]/[%D (all inputs)]
• If a 1% increase in all inputs results in a greater than 1%
increase in output, then the production function
exhibits increasing returns to scale.
•If a 1% increase in all inputs results in exactly a 1%
increase in output, then the production function
exhibits constant returns to scale.
• If a 1% increase in all inputs results in a less than 1%
increase in output, then the production function
exhibits decreasing returns to scale.
44
Returns to Scale
K
2K
Q = Q1
K
Q = Q0
0
L
2L
L
45
Returns to Scale
Returns to Scale
• It is possible for a production function to exhibit
constant returns to scale for some levels of input
usage and increasing or decreasing returns for
other levels
– the degree of returns to scale is generally
defined within a fairly narrow range of
variation in input usage
Constant Returns to Scale
• Constant returns-to-scale production functions
are homogeneous of degree one in inputs
f(tk,tl) = t1f(k,l) = tq
• The marginal productivity functions are
homogeneous of degree zero
– if a function is homogeneous of degree k, its
derivatives are homogeneous of degree k-1
Constant Returns to Scale
• Along a ray from the origin (constant K/L),
the MRTS will be the same on all isoquants
k per period
The isoquants are equally
spaced as output expands
q=3
q=2
q=1
l per period
Returns to Scale
• Returns to scale can be generalized to a
production function with n inputs
q = f(x1,x2,…,xn)
• If all inputs are multiplied by a positive
constant t, we have
f(tx1,tx2,…,txn) = tkf(x1,x2,…,xn)=tkq
– If k = 1, we have constant returns to scale
– If k < 1, we have decreasing returns to scale
– If k > 1, we have increasing returns to scale
The Linear Production Function
• Suppose that the production function is
q = f(k,l) = ak + bl
• This production function exhibits constant
returns to scale
f(tk,tl) = atk + btl = t(ak + bl) = tf(k,l)
• All isoquants are straight lines
The Linear Production Function
Capital and labor are perfect substitutes
k per period
MRTS is constant as K/L changes
slope = - b/a
q1
q2
=
q3
l per period
Fixed Proportions
• Suppose that the production function is
q = min (ak,bl) a,b > 0
• Capital and labor must always be used in a fixed
ratio
– the firm will always operate along a ray where
K/L is constant
• Because K/L is constant,  = 0
Fixed Proportions
No substitution between labor and capital is
possible
K/L is fixed at b/a
k per period
=0
q3
q3/a
q2
q1
q3/b
l per period
Cobb-Douglas Production Function
• Suppose that the production function is
q = f(k,l) = Akalb A,a,b > 0
• This production function can exhibit any returns
to scale
f(tk,tl) = A(tk)a(tl)b = Ata+b kalb = ta+bf(k,l)
– if a + b = 1  constant returns to scale
– if a + b > 1  increasing returns to scale
– if a + b < 1  decreasing returns to scale
Cobb-Douglas Production Function
• The Cobb-Douglas production function is linear
in logarithms
ln q = ln A + a ln k + b ln l
– a is the elasticity of output with respect to k
– b is the elasticity of output with respect to l
CES Production Function
• Suppose that the production function is
q = f(k,l) = [k + l] /   1,   0,  > 0
–  > 1  increasing returns to scale
–  < 1  decreasing returns to scale
• For this production function
 = 1/(1-)
–  = 1  linear production function
–  = -  fixed proportions production function
–  = 0  Cobb-Douglas production function
A Generalized Leontief Production
Function: A special case
• Suppose that the production function is
q = f(k,l) = k + l + 2(kl)0.5
• Marginal productivities are
fk = 1 + (k/l)-0.5
fl = 1 + (k/l)0.5
• Thus,
f l 1  (k / l ) 0.5
MRTS 

f k 1  (k / l ) 0.5
A Generalized Leontief Production
Function: A special case
• This function has a CES form with  = 0.5
and  = 1
• The elasticity of substitution is
1
1


2
1   0 .5
Technological Progress
Definition:
• Technological progress (or invention)
• Shifts the production function by
allowing the firm
•to achieve more output from a given
combination of inputs
•or the same output with fewer inputs.
60
Neutral Technological Progress
Technological
progress that
decreases the
amounts of labor
and capital
needed to
produce a given
output.
Affects MRTSK,L
61
Technological Progress
Labor saving technological progress results in
a fall in the MRTSL,K along any ray from the
origin
Capital saving technological progress results
in a rise in the MRTSL,K along any ray from the
origin.
62
Labor Saving / Capital deepening Technological Progress
Technological
progress that
causes the
marginal
product of
capital to
increase relative
to the marginal
product of labor
63
Capital Saving / Labour deepening Technological Progress
Technological
progress that
causes the
marginal product
of labor to
increase relative to
the marginal
product of capital
64
Technical Progress
• Methods of production change over time
• Following the development of superior
production techniques, the same level of output
can be produced with fewer inputs
– the isoquant shifts in
Technical Progress
• Suppose that the production function is
q = A(t)f(k,l)
where A(t) represents all influences that go
into determining q other than k and l
– changes in A over time represent technical
progress
• A is shown as a function of time (t)
• dA/dt > 0
Technical Progress
• Differentiating the production function with
respect to time we get
dq dA
df (k , l )

 f (k, l )  A 
dt
dt
dt
dq dA q
q  f dk f dl 

 

  

dt
dt A f (k , l)  k dt l dt 
• Dividing by q gives us
dq / dt dA / dt f / k dk f / l dl





q
A
f (k , l ) dt f (k , l ) dt
dq / dt dA / dt f
k
dk / dt f
l
dl / dt




 

q
A
k f (k , l )
k
l f (k , l )
l
Technical Progress
• For any variable x, [(dx/dt)/x] is the
proportional growth rate in x
– denote this by Gx
• Then, we can write the equation in terms of
growth rates
f
k
f
l
Gq  GA 

 Gk  
 Gl
k f (k , l )
l f (k , l )
Technical Progress
• Since
f
k
q k


  eq,k
k f (k , l ) k q
f
l
q l


  eq,l
l f (k , l ) l q
Gq  GA  eq,kGk  eq,lGl