Chapter 7
Production functions
Turning inputs into outputs.
Institutions that do this are called
firms.
Production Function
• It shows the relationship between inputs and outputs,
assuming a given technology that combines these
inputs to produce the outputs.
q  f (K, L, M,....)
cap, lab, management, etc.
• The production function shows the maximum output
attainable with given levels of the inputs and
q
technology.
L|K0
• A production function allows many different
combinations of inputs to achieve any given level of
output.
For ease of exposition, reduce the size of the
production function to two inputs and one output (one
variable input and one fixed input), q = f(K0,L).
Marginal Product
q
where K is fixed.
MPL 
 fL  0
L
•The marginal product of L is the additional output
produced by an additional unit of L with all other
inputs and technology fixed.
 MP L
 2q

 f LL  0
2
•MP is diminishing as more
L
L
and more of one input is added q
So, fL > 0 and fLL < 0 in
the relevant range of
with all other inputs and
efficient production.
technology fixed.
L|K0
q
Tangency, MPL = APL. Slope
of chord = slope of f(K0, L).
MPL is the derivative (slope) of the TPL
curve. It is maximized at the TPL
inflection point (L*) and is 0 at
Chord
Inflection
point
q = f(K0,L)
= TPL=
total
physical
product
0
MPL
APL
L|K0
Stage I
Stage II Stage III
MP L 
0
L* L**
q
L
L***
Physical efficiency of L is increasing
when q/L is increasing. Produce at
some level of L after q/L begins to
decline (L** or greater).
APL 
q
L
L|K0
maximum TPL (L***).
APL is the slope of a chord from the
origin to any point on the TPL curve.
APL is maximized at L** and declines
thereafter, but never reaches zero,
except where TPL = 0. APL rises so
long as MPL is above it and declines
when MPL is below it. APL is
maximized where the slope of the chord
is maximum. At maximum chord slope,
the slope of the TPL curve (MPL) equals
the slope of the chord (APL). This
maximum occurs at L** where the chord
is tangent to the TPL curve. Production
will always occur in Stage II between
L** and L***. Up to L** , MPL > APL
and APL is rising. Beyond L***, MPL is
negative.
Isoquant Approach
An isoquant shows various combinations of K and L that
result in a fixed level of q.
K
q3 q1 = f(K, L) is an isoquant. To find the
q2 equation for an isoquant, solve for K in
q1 terms of L and q1, where q1 is held constant.
0
L
The Marginal Rate of Technical Substitution (RTSLforK) shows
how one input (L) can be substituted for the other (K) with q
constant. It is the negative of the slope of the isoquant.
dK
RTSLK  
 0,
dL q
because dK/dL < 0.
dK

dL
• The RTS LK
is equal to the ratio of the marginal
MPL
productivities of L and K =
.
MPK
• Take the totally differential of the production function,
q  f(K, L).
f
f
dq 
 dK   dL  MPK  dK  MPL  dL
K
L
Along an isoquant dq = 0, so MPK  dK  MPL  dL
dK
Therefore, 
dL
q
MPL

 RTS LK
MPK
Characteristics of Isoquants
• Slopes of isoquants are negative in the relevant
range (Stage II; between L** and L***).
• Isoquants are strictly convex and form a strictly
convex set. Thus, RTS is diminishing. At
small L and large K, RTSLK is large (MPL is
large relative to MPK). At large L and small K,
RTS is small (MPL is small relative to MPK).
K
0
MPL is large and MPK is small.
MPL is small and MPK is large.
q1
L
q
L|K0
An increase in L causes MPL = fL to decrease (fLL<0).
A decrease in K causes MPK = fK to increase (fKK<0).
• For isoquants to be strictly convex, RTSLK must fall as L
 MPL 
increases. So, look at
dq = 0 along
 d f L 
d
MPK 
fK 
dRTS
 
 
 0 an isoquant.
dL
dL
dL
• Take the total derivative of RTSLK with respect to L using the
Quotient Rule:
 
dK  
dK 
Derivative of an
implicit function
•
•
•
•
•
fK  fLL  fLK    fL  fKL  fKK  
dL  
dL 
dRTS  

dL
fK2
If numerator <0, strictly quasiconcave function so isoquants are
Young’s Theorem
strictly convex.
dK
fL


and fKL  fLK,
Because dL
fK


dRTS fK2fLL  2fKfLfKL  fL2fKK

0
3
dL
fK
If we assume Stage II, denominator will be > 0 (fK > 0). Also assume fL > 0.
If we assume that fLL and fKK < 0 and that fKL is positive, then numerator will < 0.
dRTS
Thus, dL  0, so in Stage II isoquants are strictly convex and RTS is
diminishing.
Each of the assumptions, upon examination, makes real-world sense.
Returns to Scale
• If q = f(K,L) and K and L increase by a constant
greater than zero (m > 0), what happens to q?
• If: f(mK, mL) = mf(K,L) = mq  constant returns
f(mK, mL) < mf(K,L) = mq  decreasing returns
f(mK,mL) > mf(K,L) = mq  increasing returns
ΔKs are equal and ΔLs are equal.
K
K
K
q5
q5
q4 ΔK
ΔK
q4
q3
q3
q1
ΔL
Constant
q2
L
ΔK
q
q1 2
ΔL
q1
L
Decreasing
ΔL
q2
q3
q5
q4
L
Increasing
A constant returns production function is “homogeneous of degree
1” (linear homogeneous). That is:
f(mK, mL) = m1f(K,L) = m1q.
Constant returns production functions possess the property that
the RTS depends only on the ratio K to L, not on the level of
output (scale of production). If we move out on a ray from the
origin, K/L is constant and the RTS is constant at the point where
the ray passes through each isoquant!
Constant returns production functions are also homothetic.
However, not all homothetic production functions are constant
returns. Homothetic production functions have isoquants that are
“radial extensions” of the unit isoquant. That is, the RTS is the
same at all isoquants where they intersect any ray from the origin.
K
RTSLK is the same; therefore, the
production function is homothetic,
but increasing returns to scale.
L
• In General if: q = f(x1, x2 ,…, xn) then if all inputs
are multiplied by a positive constant, f(mx1, mx2,
…, mxn) = mkf(x1, x2, …, xn) = mkq.
• If k = 1 have constant returns to scale and the
production function is linear homogeneous.
• If k > 1 have increasing returns to scale.
• If k < 1 have decreasing returns to scale.
• True changes in scale may not actually occur in
the real world. It is a difficult management
problem to increase all inputs by the same
proportion.
Elasticity of Substitution
 L  σ  dK L  RTS  0
%Δ K
% RTS
Measures the
proportionate change
in K/L relative to a
change in RTS along a
given isoquant.
dRTS K
L
Because RTS and
K/L move in the
same direction.
(-) slope = -dK/dL = RTSLK
K
K

L
K
A
'
L'
As RTS gets smaller
in going from A to
B, K/L gets smaller.

B
q0
(-) slope= -dK/dL = RTSLK
L
• If K/L does not change much as RTSLK changes, the
two inputs are not easily substituted in production.
• If K/L changes rapidly as RTSLK changes, the two
inputs are good substitutes.
•  tells us about the shape of the isoquant.
K
e.g., different sources of N.

0
A change in RTS along an
isoquant causes a relatively large
change in K/L.  is large, so K
and L are good substitutes.
e.g., machinery and labor.
K


L
A change in RTS along
an isoquant causes a
relatively small change
in K/L.  is small, so K
and L are not good
substitutes.
0
L
• changes along most isoquant and as output moves
from one isoquant to another.
Elasticities of production
Elasticity of Production for K,
Elasticity of Production for L,
%q q K

 .
%K K q
%q q L

 .
%L L q
These are elasticities of production for K and L (eqK , eqL ).
Common Production Functions
• Linear Function
q = f(K,L) = aK + bL  K = q/a – (b/a)L is the
isoquant.
Constant returns to scale (linear homogeneous). K
 = infinity because all isoquants are straight
0
 lines. The slope of the isoquants is – b/a.
Isoquants
L
• Fixed Proportions
K
q = min(aK, bL), a and b >0.
300
No substitutability.
200
100
Slope
=
K/L
=
b/a.
 = 0 at vertices.
0
L
Isoquants are L-shaped.
Logical production is always along the ray where
K/L = b/a.
b/a is the slope of the ray that originates at 0. (The
larger “a” is the closer the isoquant is the L axis.)



When both inputs are fully utilized and
d ( K L ) RTS


 0 there is no unused surplus of either input,
dRTS
(K L )
then aK = bL and K/L = b/a.
• Cobb-Douglas, q  AK L ; A, a , b  0,   1
This function can exhibit different returns to scale
depending on the values of a and b.
a
b
• If all inputs are increased by a multiple of m then
q  A(mK)a (mL)b  Am a  b K a Lb  m a  bq
• If a + b = 1
constant returns, q increases by m
• If a + b > 1
increasing returns, q increases by ma+b
• If a + b < 1
decreasing returns, q increases by ma+b
Elasticities of production
% q q K
“a” is the Elasticity of Production for K,

  a.
“b” is the Elasticity of Production for L,
% K K q
% q q L

  b.
% L L q
These are elasticities of production for K and L (eqK , eqL ).
• Constant Elasticity of Substitution (CES)
General production function with regard to Elasticity of Substitution
and returns to scale.  is constant along an isoquant.


ρ γ/ρ 0  β  1,      1, ρ  0, γ  0
q  βK  (1  β)L
ρ
β is the distribution parameter, which determines the relative
importance of K and L.
ρ is the substitution parameter, which determines substitution
between K and L. Higher ρ means higher  = 1/(1-ρ).
γ is the scale parameter, γ  1  constant returns to scale
γ  1  increasing returns to scale
γ  1  decreasing returns to scale
Returns to Scale

q  m  βK
q  β(mK) ρ  (1  β)(mL) ρ
ρ
γ
ρ
ρ
 (1  β)Lρ


γ
γ
ρ
ρ
q  m f(K, L)
γ
γ < 1, decreasing; γ = 1, constant; γ > 1, increasing
The CES production function incorporates the linear, fixed
proportion, and Cobb Douglas production functions as special
cases. For this function,  = 1/(1- ρ).
σ  1/(1  ρ)    linear
lim ρ1
σ  1/(1  ρ)  0  fixed proportion s
lim ρ  - 
σ  1/(1  ρ)  1  Cobb - Douglas
lim ρ0
Technical Progress
• Technical progress shifts the q0 isoquant
toward the origin. That is, the firm can now
produce the same level of output with fewer
inputs, given input prices.
If the firm used L1 and K2 to produce q0t0 in period t0 at A, it
can now use L1 and K1 to produce q0t1 in period t1 at C. If it
used K1 and L2 to produce q0t0 in period t0 at B, it need only
use K1 and L1 to produce q0t1 in period t1 at point C.
K
K2
K1
A D
B
C
 
q 0t 0
q 0t 1
L1
L2
L
Labor Productivity (q/L)
q 0t1
q 0t 0
at B, with K not changing  technical
L1
L 2 progress;
q 0t
q 0t
at B, but K changes  no technical

At A, L
L2
1
progress, just substitution of L for K.
At C,
0

0
• Labor productivity (q/L) does not necessarily measure technical progress.
Going from B to C, q0t1/L1 is greater than q0t0/L2 (K unchanged), so
technical progress has been made. On the other hand, at A, q0t0/L1 is the
same as q0t1/L1 at C; technical progress has still been made because K has
declined. q/L must increase over time, holding K constant, to show
technical progress with q/L. Labor productivity is not a good measure
of technical progress because one does not know whether changes (or
not) in q/L represents going from A to B, A to C, B to C, D to C, or
some other combination.
Measuring Technical Progress
• If we use the following production function, q(t) =
A(t)f(K(t),L(t)) and let K(t) and L(t) be conventionally
defined inputs. q(t) can increase over time by adding
more to K and/or L, or it can increase over time with
A(t), holding K and L constant. The latter measures
technical progress.
• If we differentiate this function with respect to time (t),
we get: Gq  GA  eqKGK  eqLGL
• Where growth in output (Gq) over time equals growth
in output from technical progress (GA) plus growth in
K over time (GK) times the elasticity of production for
K (eqK) plus the growth in L over time (GL) times the
elasticity of production for L (eqL).
• Where
dq
dt  the rate of growth in q per unit of
q
Gq 
time.
GA 
dA
GK 
e qK
dt  rate of technical progress per unit of
A
time.
dK
dt ,
K
q K

 ,
K q
GL 
e qL
dL
dt  rates of input growth
L
per unit of time.
q L

 elasticities of production
L q
for K and L, respectively.
Example of estimating technical progress
• If labor hours increase at a rate of 2% per
year and capital services increase by 2.5%
per year and the elasticities of production
for capital and labor are 0.5 and 0.7,
respectively, and we know the rate of
growth in output to be 4% per year, what is
the rate of growth in technical progress?
4.00  G A  .5(2.5)  .7(2) Solve for GA.
G A  4.00  1.25  1.40  1.35%
= growth in total output per year
from technical progress.
Study the Cobb-Douglas example in the text with various exponents.