The Cobb-Douglas Production Function, Costs, Factor
Quantity Employed and Parametric Estimation: A Firm
Level Perspective
XIANBAI LI
School of Management and Economics
Beijing Institute of Technology, 100081 Beijing
Email: [email protected] or [email protected]
Abstract:
This paper studies the theoretically internal relationships among the
Cobb-Douglas production function, the long-run costs, the short-run costs, and
their figures. If the production characteristics of a firm can be represented by
the Cobb-Douglas production function, and if the market demand of the firm
product can be described by linear demand, we can know the long-run
equilibrium production level, the short-run equilibrium production level and the
factor quantity employed. In addition, this paper points out differences of
estimating the parameters
α
and
α
i
in the Cobb-Douglas production
function.
Key words: Cobb-Douglas production function, the long-run cost, the short-run
cost, factor quantity employed, consistent estimation
JEL classification: D24; C13
1
1. Introduction
The Cobb-Douglas production function plays an important role in economic
research. It can be used to study the impact on the agricultural output of the
education and the public expenditures on agricultural research and extension
[Zvi Griliches, 1964], to study the output elasticity of some input [M. S.
Feldstein, 1967], to study whether the national and international aid to the
agricultural sector of less developed countries has made an identifiable
contribution [Wayne Schutjer and Dale Weigel, 1969], to study allocation
efficiency in a developing agricultural economy [Yukon Huang, 1971], to study
the measurement of total factor input, technical change and output [Rolf
Krengel, 1972], to assess the relative roles of stages of economic development
and explain the productivity growth differences in agriculture across countries
[Erkin I. Bairam and Shaun D. Mcrae, 1999], to study the role of education in
economic growth [Namchul Lee, 2000], to explain and compare the economic
growth in different areas or countries [Gregory Chow, 2002], to study the
impact of computerization on firm productivity [Y. C. Ng and M. K. Chang,
2003], to study the relationship between the policy measures on innovation and
the R&D performance [Kuen-Hung Tsai and Jiann-Chyuan Wang, 2004], to
compare macroeconomic returns on human and public capital [Alvaro Manuel
Pina and Miguel St. Aubyn, 2005], or to study firms’ productivity and R&D
spillovers [Michele Cincera, 2005]. Much of the research mainly focuses on the
fields of macroeconomics and industrial organization and on the relationships
2
between inputs and the output. Although there is a considerable amount of
research related to the Cobb-Douglas production function, the relationships
between the C-D production function and costs at the firm level are not studied
sufficiently. Based on the C-D production function with
k inputs, this paper
systematically shows the theoretically internal relations among the long-run
cost, the short-run cost, the quantity of factors hired and the cost curve graphs.
This paper is organized in the following manner. The long-run costs and their
curves, including the equilibrium quantity of factors employed, are studied in
the second section. The short-run costs and their curves, including the
equilibrium quantity of factors employed, are studied in the third section. The
estimation and the rate of some parameter is presented and interpreted in the
fourth section. The conclusions are given in the final section.
2. The long-run cost, cost curve graphs and the long-run
equilibrium quantity of factors employed
The Cobb-Douglas production function with
k
inputs is
y = α xα xα L xα ，α > 0 ，α > 0 ，i = 1, 2,L, k . Because the long-run
1
2
1
2
k
i
k
y
cost with production quantity
is the minimum cost when producing
y , our
k
objective function is
min ∑ p x . In order to ascertain the equilibrium
i
i
i =1
quantity of factors employed, construct the Lagrange function
L = ∑ p x + λ [ y − α xα xα L xα ]. The necessary conditions of
k
1
i =1
i
i
1
2
2
k
k
3
∂L
= 0, i = 1, 2,L, k . Solving these equations we
∂x
px px
p
=
=L= x
(1)
minimizing cost are
i
1
1
get
α
2
2
α
1
k
k
α
2
k
Using (1) to simplify the Cobb-Douglas production function gives
⎛px α
y = α xα ⎜⎜
⋅
⎝ α p
1
1
α
1
2
1
1
α +α
⎛p⎞
=α⎜ ⎟
⎝α ⎠
1
2
α
+L+
1
1
= A xB
k
k
2
⎞
⎟⎟
⎠
⎛α
⎜⎜
⎝p
2
3
α
2
1
1
1
1
1
⎞
⎟⎟
⎠
1
⎛α
⎜⎜
⎝p
3
α
2
⎞
⎛p
⎟⎟ L ⎜⎜ x ⋅α
⎠
⎝ α p
α
⎞
⎟⎟
⎠
3
1
1
1
α
⎛ ⎞
⎞
⎟⎟ L⎜⎜ α ⎟⎟ xα
⎠
⎝p ⎠
k
k
k
k
k
α
+
1
2
α
+L+
k
1
2
k
( 2)
k
1
α +α
⎛p⎞
where， A = α ⎜ ⎟
⎝α ⎠
1
2
α
+L+
k
1
k
1
B = α +α +L+α
k
α
⎛px α
⎜⎜
⋅
⎝ α p
1
2
k
⎛α
⎜⎜
⎝p
α
1
1
⎞
⎟⎟
⎠
1
⎛α
⎜⎜
⎝p
2
2
α
2
k
k
，
k
.
⎛ y
=
⎜
⎟
x
⎝A ⎠
1
⎞ Bk
From (2) we have
α
⎛ ⎞
⎞
⎟⎟ L⎜⎜ α ⎟⎟
⎠
⎝p ⎠
(3)
1
k
i
From (1) and (3) we get the
th
input quantity
1
p ⎛ y ⎞B
α
⎜ ⎟ , i = 1, 2,L, k
x=
pα ⎝A ⎠
k
i
1
i
1
( 4)
i
k
Using (4)，we obtain the long-run total cost function
⎡
p ⎛ y ⎞B ⎤
α
LTC = ∑ p x = ∑ ⎢ p ⋅
⎜ ⎟ ⎥
pα ⎝A ⎠ ⎥
⎢⎣
⎦
1
k
i =1
k
i
i
i =1
k
i
1
i
1
i
k
4
1
1
p ⎛ y ⎞B
p B ⎛ y ⎞B
= ⎜ ⎟ ⋅ ∑α =
⎜ ⎟
α ⎝A ⎠
α ⎝A ⎠
k
k
1
1
k
k
1
(5)
i
i =1
1
k
k
From (5) we get the long-run average cost
LAC =
pB
1
α
k
−1
k
A
Bk
y
( 1−
Bk ) Bk
( 6)
1
From (5) we also get the long-run marginal cost
1
p ⎛ y ⎞B
LMC = LTC ′ =
⎜ ⎟
α A ⎝A ⎠
−1
k
( 7)
1
1
k
k
LTC , LAC , LMC , we discuss their
In order to get the curve graphs of
natures as follows.
p (1 − B ) ⎛ y
From (7) we get LMC ′ = LTC ′′ =
⎜ ⎟
α A B ⎝A ⎠
If B < 1，then LMC ′ = LTC ′′ > 0 ；if B > 1，then
k
1
2
k
1
k
k
1
⎞ Bk
−2
(8)
k
k
LMC ′ = LTC ′′ < 0 .
1
−3
p (1 − B ) (1 − 2 B ) ⎛ y
From (8) we get LMC ′′ =
⎜ ⎟
α A B
B
⎝A ⎠
1
1
or B > 1，then LMC ′′ > 0 ；if
If B <
< B < 1 ，then
2
2
LMC ′′ < 0 .
k
1
3
k
1
k
If
k
k
k
( 9)
k
k
From (6) we obtain
⎞ Bk
k
LAC ′ =
p (1 − B )
k
1
α
−1
k
A
Bk
y
( 1−2
Bk ) Bk
(10)
1
B < 1，then LAC ′ > 0 ；if B > 1，then LAC ′ < 0 .
k
k
From (10) we get
5
p (1 − B )(1 − 2 B ) B
B B
(11)
A y
α
B
1
1
If B <
or B > 1，then LAC ′′ > 0 ；if
< B < 1 ，then
2
2
LAC ′′ < 0 .
LAC ′′ =
k
1
−1
k
k
( 1− 3
k
k
)
k
k
1
k
k
k
1
1 ⎞
pB⎛ y
(12)
⎜ ⎟ ⎜ − 1⎟
y
α ⎝A ⎠ ⎝
⎠
y > 1，then LAC < LTC ；if y = 1，
LAC − LTC =
From (5) (6) we have
⎞Bk ⎛
k
1
1
If
y < 1，then LAC > LTC ；if
LAC = LTC ，i.e. LAC
then
LTC
and
k
intersect at
y = 1.
1
LMC − LTC =
From (5) (7) we have
1
⎞
p⎛ y
(13)
⎜ ⎟ ⎜ −B ⎟
y
α ⎝A ⎠ ⎝
⎠
1
y > ，then LMC < LTC ；if
B
1
k
1
If
1
y<
y=
B
1
B
，then
LMC > LTC ；if
k
，then
⎞ Bk ⎛
k
k
LMC
LTC
and
intersect.
k
From (6) and (7) we have
1
⎞ Bk
p⎛1
LAC − LMC = ⎜ ⎟ y B (B − 1)
α ⎝A ⎠
If B > 1，then LAC > LMC ; if B > 1
1
1
−1
k
1
(14)
k
k
k
k
and
y → ∞ ，then
( LAC − LMC ) → 0 ；if B < 1，then LAC < LMC .
k
Based on the conclusions of (8) through (14)，we discuss the curve graphs of
following five situations
2
(I) If
1
py
LTC
,
then
=
=
B
2
2α A
， LAC
2
1
k
1
k
=
py
2α A
1
1
， LMC
2
k
=
py
α A
1
1
2
k
.
6
The cost curves are shown in Figure 1.
B = 1 ，then LTC =
(II) If
k
py
α A
1
1
,
LAC = LMC =
k
p
α A
1
1
. The cost
k
curves are shown in Figure 2.
(III) If
B > 1，then LTC ′ > 0 ，LTC ′′ < 0 ，LMC ′ < 0 ，LMC ′′ > 0 ，
k
LAC ′ < 0 ， LAC ′′ > 0 . The cost curves are shown in Figure 3.
1
< B < 1 ， then LTC ′ > 0 ， LTC ′′ > 0 ， LMC ′ > 0 ，
2
LMC ′′ < 0 ，LAC ′ > 0 ，LAC ′′ < 0 . The cost curves are shown in Figure
(IV) If
k
4.
1
，then LTC ′ > 0 ，LTC ′′ > 0 ，LMC ′ > 0 ，LMC ′′ > 0 ，
2
LAC ′ > 0 ， LAC ′′ > 0 . The cost curves are shown in Figure 5.
B <
(V) If
k
LTC
LTC
LMC
LAC=LMC
LAC
y
1
y
Fig 1
2
Fig 2
1
LTC
LMC
LTC
LMC
LAC
LAC
y
1B
1
k
Fig 3
Fig4
1
1B
k
2
y
7
LTC
LMC
LAC
y
Fig 5
2
1
1
B
k
If the market demand facing the firm is linear
then the total revenue is
p = g − hy ，g > 0 ，h > 0 ，
TR = py = gy − h y
2
，and the marginal revenue is
MR = g − 2hy . The long-run equilibrium production quantity y
satisfy
MR = LMC ，i.e., g − 2h y =
*
p
α A
1
1
After knowing
y
*
k
⎛y
⎜⎜ ⎟⎟
⎝A ⎠
*
1
⎞ Bk
*
should
−1
(15)
k
，from（4）we can get all the variable input quantity
employed when producing
y
*
.
3．The short-run cost, curve graphs and the equilibrium
inputs employed
The short-run equilibrium production quantity and the equilibrium inputs
employed are relatively complicated, because they are decided not only by the
number of fixed factors but also by the kind of fixed factors. The following
discussion is about the short-run costs with
fixed factors, where
j
j
variable factors or ( k
− j)
j = 1, 2,L, k − 1. We can reasonably assume that the
variable factors are
x ，x
1
2
，…， x j respectively, and that the ( k
− j)
8
x
fixed factors are
j +1
j
，x j +2 ，…，xk respectively. As to
(
j
k
)
variable factors, the
min ∑ p x + ∑ p x . Construct the Lagrange function
objective function is
i
i
i =1
i
i
i = j +1
L = ∑ p x + ∑ p x + λ ( y − α xα L xα xα L xα )
j
k
1
i
i
i =1
i
i
i = j +1
j +1
j
k
j +1
j
1
k
∂L
= 0 , (i = 1, 2, L, j)，we have
∂x
From the first order condition
i
p x
px px
=
=L=
1
1
α
2
2
α
1
j
α
2
j
(16)
j
Using (16) to simplify the Cobb-Douglas production function gives
⎛px α
α
⋅
y = α x ⎜⎜
⎝ α p
1
1
1
α
⎞
⎟⎟
⎠
2
1
1
α
B
⎛ p ⎞ ⎛α ⎞
= α ⎜ ⎟ ⎜⎜ ⎟⎟
⎝α ⎠ ⎝ p ⎠
= A x B xα L xα
j
1
1
1
1
j +1
j
j
1
j +1
1
2
⎛α
⎜⎜
⎝p
2
⎛px α
⎜⎜
⋅
⎝ α p
1
1
α
2
1
3
α
2
j
⎞ α
⎟ x L xα
⎟
⎠
j
j +1
j +1
j
k
k
j
j +1
j
j +1
1
2
1
1
j
k
k
j
(17)
k
k
⎛ p ⎞ ⎛α
where， A = α ⎜ ⎟ ⎜
⎜
⎝α ⎠ ⎝ p
B = α +α +L+α
α
1
1
1
1
j
1
3
α
3
⎛ ⎞
⎞
⎟⎟ L⎜ α ⎟ x B xα L xα
⎜ ⎟
⎠
⎝p ⎠
Bj
j
1
α
⎛px α
⎞
⎟⎟ L⎜
⎜ α ⋅p
⎠
⎝
2
⎞
⎟⎟
⎠
1
⎛α
⎜⎜
⎝p
α
2
2
α
⎛ ⎞
⎞
⎟⎟ L⎜ α ⎟
⎜p ⎟
⎠
⎝ ⎠
2
j
j
，
j
j
From (17) we obtain
⎛
y
⎟
=
x ⎜⎜
α L α ⎟
x ⎠
⎝Ax
1
⎞B j
1
j
j +1
j +1
(18)
k
k
From (16) and (18) we get the input quantity of all kinds of variable factors
α p =α p
x=
x
pα
pα
i
i
1
1
1
i
i
1
i
1
⎛
y
⎜⎜
α
α
⎝ A x Lx
j
j +1
j +1
k
1
⎞B
⎟⎟ , i = 1, 2,L, j
⎠
j
k
(19)
9
Using (19) and noticing that
x
j +1
x
, …,
k
are fixed factors, we get the
short-run total cost
j
STC = ∑ p x + p x + L + p x
i
i =1
j +1
j +1
i
k
⎡
y
p⎛
= ∑ ⎢ p ⋅ α ⎜⎜
p α ⎝ A xα L xα
⎢
⎣
j
i
i =1
i
⎞B j ⎤
1
1
i
1
j +1
j +1
j
k
⎟⎟ ⎥ + p x + L + p x
⎠ ⎥⎦
j +1
j +1
k
k
k
k
1
y
pB ⎛
⎟ + p x +L+ p x
⎜⎜
( 20)
α L α ⎟
x ⎠
α ⎝Ax
As to (20), what are the values of x , …, x to minimize STC when
=
⎞B j
j
1
1
j +1
j +1
j +1
j
j +1
k
k
j +1
producing the production quantity
k
y ? This is a question of optimum quantities
of fixed factors when producing the production quantity
x
k
k
k
y . Actually, x
, …,
should be seen as parametric variables when answering this question.
∂STC
∂STC
= 0 , …,
= 0 , i.e.,
∂x
∂x
− j ) equations:
Therefore, we have ( k
j +1
y
p⎛
⎜⎜
⎟
α L α ⎟
x ⎠
α ⎝Ax
1
⎞B j
1
1
j +1
j
j +1
j +1
k
−1
y
(− α
α L α
x
Ax
j +2
k
j+2
j
k
k
j +1
) x α + p = 0 (21)
−
j +1
−1
j +1
j +1
k
………………………………………………………………………………
1
⎞B j
−1
y
y
p⎛
α + p = 0 ( 22)
⎟⎟
⎜⎜
(
)
−
x
α
α ⎝ A xα Lxα ⎠ A xα L xα
Solving the above ( k − j ) equations from (21) to (22) we get x , …, x
1
1
j
j +1
j +1
k
j +1
k
k
j
j +1
−
k
k −1
k
−1
k
k −1
j +1
which must be equal to the expression（4） when
x
j +1
, …,
x
k
k
,
i = j + 1,L, k , because
are equivalent to variable factors when they are seen as
parametric variables.
10
From (20) we have the short-run marginal cost
1
SMC = STC ′ =
⎞B j
1
p⎛
⎟ yB
⎜⎜
α L α ⎟
α ⎝Ax
x ⎠
1
1
j
j +1
j +1
1
−1
( 23)
j
k
k
If the market demand of the firm’s product is linear
h > 0 ，and if there are j
y
production quantity
*
p = g − hy ， g > 0 ，
variable factors，then the short-run equilibrium
should satisfy
MR = SMC ，i.e.,
1
1
p⎛
⎟ yB
g − 2h y = ⎜⎜
( 24)
α L α ⎟
x ⎠
α ⎝Ax
After knowing y , we can from (19) get the quantity of variable input factors
*
⎞B j
1
*
1
−1
j
1
j
j +1
j +1
k
k
*
employed when in the state of short-run equilibrium. The following is to
discuss the natures of the short-run cost curves in order to obtain the cost curve
graphs. From (23) we have
1
SMC ′ = STC ′′ =
1
⎛1
⎞
p⎛
⎟⎟ ⎜ − 1⎟ y B
⎜⎜
α ⎝ A x α L xα ⎠ ⎝ B
⎠
1
1
If
⎞B j
j +1
j +1
j
1
−2
( 25)
j
k
j
k
B < 1, then SMC ′ = STC ′′ > 0 ；if B > 1, then
j
j
SMC ′ = STC ′′ < 0 .
From (25) we obtain
1
p⎛
⎜⎜
α ⎝ A xα L xα
1
If B j > 1 or B j < , then
2
SMC ′′ < 0 .
SMC ′′ =
1
⎞B j
1
1
j
j +1
j +1
k
⎞
⎞⎛ 1
⎛1
⎟⎟ ⎜ − 1⎟⎜ − 2 ⎟ y B
( 26)
⎠
⎠⎝ B
⎠ ⎝B
1
SMC ′′ > 0 ；If < B j < 1 , then
2
1
−3
j
k
j
j
From (20) we get the short-run average cost
11
1
⎞B j
1
p ⎛
⎟ yB + ( p x + L+ p x ) y
SAC = B ⎜⎜
α L α ⎟
α ⎝Ax
x ⎠
j
1
1
−1
j
1
j
j +1
j +1
j +1
j +1
−1
k
k
( 27)
k
k
From (27) we have
1
p⎛
SAC ′ = ⎜⎜
α ⎝ A xα L xα
1
j
1
1
j +1
j +1
j
⎞B
⎟⎟ (1 − B ) y B − ( p x + L + p x ) y
⎠
1
k
k
−2
j
j
j +1
If
B > 1, then SAC ′ < 0 ；If B < 1
If
B <1
j
k
−2
k
( 28)
y < δ , then SAC ′ < 0 ；
y > δ , then SAC ′ > 0 ；
and
j
Where,
and
j
j +1
⎡
⎢
⎢
p x +L+ p x
δ =⎢
1
⎢p ⎛
⎞B
1
⎢ 1 ⎜⎜
⎟ (1− B
α L α ⎟
x ⎠
⎢⎣ α 1 ⎝ A x
j +1
j +1
k
k
j
j +1
j +1
j
k
k
j
⎤
⎥
⎥
⎥
⎥
)⎥
⎥⎦
Bj
.
From (28) we get
1
SAC ′′ =
⎞B j
1
⎛1
⎞
p⎛
⎜⎜
⎟ (1 − B )⎜ − 2 ⎟ y B + 2( p x + L + p x ) y ( 29)
α L α ⎟
α ⎝Ax
x ⎠
⎝B
⎠
1
1
If
If
If
j
B >1
or
j
j +1
j +1
B <
1
< B <1
2
1
< B <1
2
j
1
−3
−3
j
j
j +1
j +1
k
k
k
k
j
1
, then SAC ′′ > 0 ；
2
j
and
y < β , then SAC ′′ > 0 ；
j
and
y > β , then SAC ′′ < 0 ；
12
Where,
⎡
⎢
⎢
p x +L+ p x
β =⎢
1
⎢p ⎛
⎞B
⎛
1
1
⎢ 1 ⎜⎜
⎟
⎜
(
)
1
1
−
−
B
⎜ 2
α
α ⎟
B
⎝
⎣⎢ α 1 ⎝ A x Lx ⎠
j +1
j +1
k
k
j
j
j +1
j +1
j
k
k
j
⎤
⎥
⎥
⎥
⎞⎥
⎟⎟ ⎥
⎠ ⎦⎥
Bj
.
From (23) and (27) we obtain
1
p⎛
SAC − SMC = ⎜⎜
α ⎝ A xα L x α
1
j
1
1
If
j
j +1
j +1
⎞B
⎟⎟ (B − 1) y B + ( p x + L + p x ) y
⎠
1
k
k
−1
j
j
j +1
j +1
k
B > 1, then SAC > SMC ，so the two curves SAC
j
not intersect. However, if
B >1
and
j
( SAC − SMC ) → 0 , i.e., SAC
−1
k
and
(30)
SMC
do
y → ∞ , we have
and
SMC
become closer and closer
infinitely.
If
B <1
j
and
y = δ , then SAC = SMC ，i.e., the two curves intersect at
y =δ ;
If
B <1
and
y < δ , then SAC > SMC ;
If
B <1
and
y > δ , then SAC < SMC .
j
j
From (20) and (27) we obtain
⎡p
SAC − STC = ⎢ B
⎢ α
⎣
1
1
If
⎤⎛ 1 ⎞
⎛
y
⎜⎜
⎟⎟ + p x + L + p x ⎥⎜ − 1⎟ (31)
α
α
⎥⎝ y ⎠
⎝ A x Lx ⎠
⎦
1
j
⎞B j
j
j +1
j +1
j +1
j +1
k
k
k
k
y < 1, then SAC > STC ；if y > 1, then SAC < STC ；if y = 1,
then
SAC = STC ，i.e., SAC
and
STC
intersect at
y = 1.
What we discuss is just about the general natures of the short-run cost
curves, and we do not discuss the situations of both
B =1
j
and
B = 1 2.
j
13
STC 、
The following is about to discuss all the five kinds of curve graphs of
SAC 、 SMC .
p
B = 1, then STC =
(I) If
p
α A xα L xα
j +1
j
SMC =
j
j +1
j +1
j +1
j +1
j +1
k
k
k
k
k
k
k
，
1
α A x L xα
j
j +1
k
+ ( p x +L+ p x ) y ，
p
1
α
j +1
−1
1
1
α
α A x Lx
1
SAC =
y + p x +L+ p x ,
1
j
α
j +1
j +1
k
k
SAC ′ = −( p x + L + p x ) y < 0 ，
−2
j +1
j +1
k
k
SAC ′′ = 2( p x + L + p x ) y > 0 .
−3
j +1
If
j +1
k
k
y → ∞ , then SAC → SMC ，i.e., SMC
See Figure 6, Figure 7, Figure 8, where
a=
is the asymptotic line of SAC .
p
α A x L xα
1
b = p x +L+ p x
j +1
j +1
k
k
，
1
j
α
j +1
j +1
k
k
.
STC
STC
a+b
SAC
a+b
b
b
SMC
Fig 6:
a=b
1
y
SAC
SMC
a
Fig 7：a < b
1
y
14
STC
a+b
SAC
a
SMC
b
Fig 8：a > b
(II) If
y
1
1
, then
2
B =
j
p
STC =
2α
1
1
⎛
1
⎜⎜
α
α
⎝ A x Lx
j +1
j +1
j
k
2
⎞
⎟⎟ y + p x + L + p x
⎠
2
j +1
k
1
p⎛
SMC = STC ′ = ⎜⎜
α ⎝ A xα Lxα
j +1
j +1
j
k
k
p
SAC =
2α
j +1
j +1
⎛
1
⎜⎜
α
α
⎝ A x Lx
1
1
j
j +1
j +1
j
j +1
j +1
k
⎞
⎟⎟ > 0 ,
⎠
2
⎞
⎟⎟ y + ( p x + L + p x ) y ,
⎠
−1
j +1
k
j +1
k
k
2
⎞
⎟⎟ − ( p x + L + p x ) y ,
⎠
−2
1
1
k
k
k
1
p ⎛
⎜⎜
SAC ′ =
2 α ⎝ A xα L xα
,
2
1
j
k
⎞
⎟⎟ y > 0 ,
⎠
1
p⎛
SMC ′ = STC ′′ = ⎜⎜
α ⎝ A xα Lxα
1
k
2
1
1
j +1
j +1
k
j +1
k
k
SAC ′′ = 2( p x + L + p x ) y > 0 .
−3
j +1
j +1
If
k
k
y < ξ ，then SAC ′ < 0 ；If y > ξ ，then SAC ′ > 0 ；If y = ξ ，then
SAC ′ = 0
and
SMC = SAC = 2 α p ( p x + L + p x ) (α A xα L xα ).
1
1
j +1
j +1
j +1
k
k
1
j
j +1
k
k
ξ = A xα L xα 2 α ( p x + L + p x ) p . Actually, ξ
the value of δ when B = 1 2 .
Here
j +1
j
j +1
k
1
k
j +1
j +1
k
k
1
is
j
15
1
p ⎛
If y = 1，then SAC = STC =
⎜⎜
2α ⎝ A xα Lxα
2
1
1
From
p
2α
1
1
STC = SMC
⎛
1
⎜⎜
α
α
⎝ A x Lx
j
j +1
j +1
⎞
p
⎟⎟ y −
α
⎠
2
1
1
k
k
j +1
k
we get a quadratic equation of
2
k
j +1
j +1
j
⎞
⎟⎟ + ( p x + L + p x ) .
⎠
⎛
1
⎜⎜
α
α
⎝ A x Lx
j
j +1
j +1
k
k
k
y:
2
⎞
⎟⎟ y + p x + L + p x = 0 .
⎠
j +1
k
j +1
k
k
ξ < 1，then the quadratic equation has two different real roots:
Therefore, if
y =1± 1−ξ
2
. At this time, the line
SMC
and the parabola
quadratic equation has no real root, i.e., the line
do not intersect. See Figure 10. If
has the same two real roots
of the parabola
SMC
STC
ξ > 1，then the
have two different points of intersection. See Figure 9. If
STC
j +1
and the parabola
ξ = 1 , then the quadratic equation
y = 1; at this time the line SMC
is a tangent
STC . See Figure 11.
STC
STC
SMC
SMC
SAC
SAC
b
b
Fig 9
ξ
1
y
Fig 10
1
ξ
y
16
STC
SMC
SAC
STC
SAC
b
b
Fig 11
ξ =1
SMC
Fig 12
y
1
y
B > 1, then STC ′ > 0 ，STC ′′ < 0 ；SMC ′ < 0 ，SMC ′′ > 0 ；
(III) If
j
SAC ′ < 0 ， SAC ′′ > 0 ； SAC > SMC ；When y → ∞ ,
( SAC − SMC ) → 0 , i.e., SAC
and
SMC
come together infinitely.
Cost curves are shown in Figure 12.
1
, then ST C ′ > 0 ，STC ′′ > 0 ；SMC ′ > 0 ，SMC ′′ > 0 ；
2
y < δ , then SAC ′ < 0 ; If y > δ , then SAC ′ > 0 and SAC ′′ > 0 .
(IV) If
If
B <
j
Based on
δ < 1、δ = 1、δ > 1，the cost cures are shown in Figures 13, 14,
15, 16. Figures 15 and 16 are all about
δ > 1，but in Figure 15 STC
SMC
STC
do intersect, and in Figure 16,
and
SMC
and
do not intersect.
Let’s give an example to describe the situation represented in Figure 15. For
example, let
B =
j
1
1
p⎛
⎜⎜
，let
3
α ⎝ A x α Lx α
1
1
p x + L + p x = 3 ，then
j +1
j +1
3
k
k
j
j +1
j +1
k
k
⎞
⎟⎟ = 3，and let
⎠
1
3
3
δ = ⎜⎛ ⎟⎞ > 1，
⎝2⎠
STC − SMC = y − 3 y + 3 = f ( y ) ， f (1) = 1 > 0 ，
3
2
f ( 2) = −1 < 0 ， f ′( y ) = 3 y ( y − 2) . If 1 < y < 2 ，then f ′( y ) < 0 .
17
y
Therefore there exists
SMC
**
which satisfies
y=y
intersect at
f ( y ) = 0 ，i.e., STC
**
and
1
3
**
. Due to
3
1 < δ = ⎜⎛ ⎟⎞ < 2 ，and due to
⎝2⎠
f (δ ) = 4.5 − 3 × 2.25 > 0 ，so δ < y
1
3
**
. See Figure 15.
The following is another example describing the situation represented in Figure
16. For example, let
B =
j
2
，
5
let
1
p⎛
⎜⎜
α ⎝ A x α L xα
5
2
1
1
j +1
j +1
j
k
k
⎞
⎟⎟ = 6 ，and let
⎠
2
5
⎛ 11 ⎞ > 1 ，
p x + L + p x = 11，then δ = ⎜
⎟
⎝ 3.6 ⎠
j +1
j +1
k
k
STC − SMC = 2.4 y − 6 y + 11 = f ( y ) ，f ′( y ) = 3 y ( 2 y − 3) .
5
2
If
3
2
y < 1.5 , then f ′( y ) < 0 ; If y = 1.5 , then f ′( y ) = 0 ; If y > 1.5 ,
then
f ′( y ) > 0 . f (1.5) = 11 − 3.6 1.5 > 0 . Therefore, ∀y > 0 ,
we have
and
f ( y ) > 0 ，i.e., ∀y > 0 , we have STC > SMC , i.e., STC
SMC
do not intersect. See Figure 16.
STC
SMC
SAC
SAC
SMC
STC
b
b
Fig 13
δ
1
y
Fig 14
δ =1
y
18
SMC
STC
STC
SMC
b
b
SAC
δ
1
y
**
SAC
y
δ
1
Fig 15
y
Fig 16
Bj
1 ⎞
1
δ ⎛
⎟⎟ < 1, i.e., δ < β . At this time,
= ⎜⎜1−
(V) If
< B < 1 , then
2
β ⎝ 2B ⎠
ST C ′ > 0 ， STC ′′ > 0 ； SMC ′ > 0 ， SMC ′′ < 0 ；if y < δ then
j
j
SAC ′ < 0 ，if y > δ
y>β
then
then
SAC ′ > 0 ；If y < β
SAC ′′ < 0 . According to the magnitude of δ 、 β
cost curves are shown in Figures 17, 18, 19, 20, 21.
STC
STC
intersect in all of these five figures. In Figure 22,
tangential to each other, and the tangent condition is both
and
STC = SMC . From STC ′( y ) = SMC ′( y )
y =
0
0
1
0
and
and
STC = SMC
are
we have
1
B
−1
into
j
and simplifying it, we have the following tangent condition:
p x +L+ p x
⎛ 1− B
⎟⎟
= ⎜⎜
⎞B ⎝ B ⎠
1
p⎛
⎟
(1 − B ) ⎜⎜
α L α ⎟
α ⎝Ax
x ⎠
j +1
SMC
STC ′ = SMC ′
0
j
and 1, the
SMC
− 1 . Obviously, we have 0 < y < 1 . Putting y =
0
B
SAC ′′ > 0 ，if
then
j +1
k
k
j
1
⎞B j
1
j
−2
.
j
1
j
1
j
j +1
j +1
k
k
Under this tangent condition, simplifying the
δ
expression gives
19
2
B j −1
⎛
⎞
δ = ⎜⎜ B ⎟⎟ .
⎝ 1− B ⎠
1
Due to
< B < 1 , so δ > 1, i.e., If STC and SMC are tangent, there
2
exists δ > 1. See Figure 22. In addition, what we need to specially point out is
j
j
j
that if the three points of intersection of
STC 、 STC
and
SMC
SAC
SMC 、 SAC
and
become one, then
SMC
and
must be a line, and the
figure at this time is similar to Figure 11. Figure 23 shows the separation of
STC
and
SMC .
STC
STC
SMC
SMC
SAC
b
Fig 17
SAC
b
β
δ
1
y
Fig 18
β =1
δ
y
SMC
STC
STC
SMC
SAC
SAC
b
b
δ
Fig 19
1
β
y
δ =1
β
y
Fig 20
20
SMC
STC
STC
SMC
SAC
SAC
b
b
δ β
1
y
y0
1
β
δ
y
Fig 22
Fig 21
SMC
STC
SAC
b
1
β
δ
y
Fig 23
We just discuss the cost curves in the above-mentioned five conditions.
After having these cost curves we can easily draw the short-run equilibrium
figure of the firm. All in all, what we discuss previously is mainly about cost
functions and cost curves when there are
fixed factors. If
(k − j )
variable factors or
j = 1 , i.e., there is only one variable factor x , then
1
A = α ，B = α
1
1
j
1
， and then the short-run total cost function accordingly is
1
⎞α
y
⎛
STC = p ⎜ α α
⎟ + p x +L+ p x
α
L
α
⎝ x x
x ⎠
1
2
2
k
3
3
1
2
2
k
k
. At this time we can
k
21
α = 1、α =
discuss short-run cost curves according to five conditions:
1
1
1
、
2
1 1
、 < α < 1. If j = ( k − 1) , i.e., there are ( k − 1 )
2 2
variable factors x , x , …, x ，or there is only one fixed factor x ，then
α > 1 、α <
1
1
1
1
k −1
2
the short-run total cost is
k
1
pB ⎛ y
⎟
⎜
α
α ⎝A x ⎠
STC =
⎞ Bk −1
k −1
1
k −1
1
+p x
k
k
k
. At this time
k
we can discuss the cost curves based on the following five conditions:
B = 1、 B =
k −1
k −1
1
1 1
、 B > 1、 B < 、 < B < 1 .
2
2 2
k −1
k −1
k −1
4. Parameter Estimation
α
Here we just emphasize the estimation of the parameter
rate of the parameter
and the growth
α . In order to estimate the parameters in the
Cobb-Douglas production function, consider an error term and rewrite the
y = α xα xα L xα e ，i.e.,
production function as
1
2
1
k
u
k
2
ln y = ln α + α ln x + α ln x + L + α ln x + u
1
Let
then
1
ln y = Y , ln α = α
0
,
2
k
2
ln x = X
1
1
ln x = X
,
2
2
ln x = X
, …,
k
k
,
Y = α + α X + α X + L + α ln x + u
0
1
1
2
2
k
k
Under some conditions we can get the OLS estimators
Let
k
αˆ = eαˆ
0
. Let’s consider the nature of
αˆ , αˆ , αˆ ,L, αˆ .
0
1
2
k
α̂ .
(32) , then
αˆ ~ N ( μ , σ )
1
⎡ (x−μ ) ⎤
E (αˆ ) = E (eαˆ ) =
dx = exp⎛⎜ μ + σ ⎞⎟ (33)
∫ e exp ⎢ −
⎥
2π σ
2⎠
⎝
⎣ 2σ ⎦
If the sampling distribution of
α̂
0
is
2
0
2
+∞
0
−∞
2
x
2
22
1
⎡ (x − μ ) ⎤
E (αˆ ) = E (e αˆ ) =
exp
e
∫
⎢ − 2 σ ⎥dx
2π σ
⎦
⎣
2
2
+∞
2
2x
0
−∞
2
= exp(2 μ + 2 σ
2
)
(34)
From (33) and (34) we have
Var (αˆ ) = E (αˆ ) − [E (αˆ )] = exp(2 μ )[exp(2σ ) − exp(σ )] (35)
2
2
If
α̂
0
2
α
is a consistent estimator of
lim E (αˆ ) = lim μ = α
0
n →∞
n→∞
2
n →∞
, then we have both (36) and (37)：
(36)
0
limVar (αˆ ) = lim σ = 0
0
0
2
(37)
n→∞
At this time，from (33), (36) and (37) we obtain
⎛
σ
lim E (αˆ ) = lim exp⎜ μ +
2
⎝
2
n →∞
n→∞
⎞
⎟ = eα = α
⎠
(38)
0
From (35), (36) and (37) we obtain
μ
limVar (αˆ ) = lim {e [exp( 2σ ) − exp(σ )]} = 0
2
n →∞
2
(39)
n →∞
From (38) and (39) we know that
say, if (32) is correct and if
consistent estimator of
If
2
E (αˆ ) = μ = α
0
obtaining from
0
α̂
α̂
α . That is to
is a consistent estimator of
0
is a consistent estimator of
α̂
0
α
0
, then
α̂
is a
α.
，i.e., if
αˆ = eαˆ
is an unbiased estimator of
is not unbiased estimator of
0
α
0
, then the
α̂
α . This is because
E (αˆ ) = E (eαˆ ) ≠ eE (αˆ ) = eα = α . From (33) we know
0
0
E (αˆ ) = exp ⎜⎛ α + σ
2
⎝
2
0
0
⎞
⎛σ ⎞
⎟ = α exp ⎜ ⎟ ( 40)
⎠
⎝2⎠
2
23
exp(αˆ )
exp(σ 2 )
0
From (40) we know that it is
2
α
that is an unbiased estimator of
under the assumption (32).
In addition, what we need to point out is that
α
constant when estimating them.
α , α , α ,L, α
1
k
2
are
denotes the ‘technology level’, specifically
including management level, management system, technology advances and the
like. Generally speaking technology level doesn’t change during a short period.
If we want to estimate the influence of technology advance on the output, then
α
α
changes with time, i.e.,
α , α ,L, α
1
2
k
is a function of time
t . However,
are still constant. Rewriting the production function as
y = α (t ) x (t )α x (t )α L x (t )α
1
1
2
k
we have
k
2
d dt
d dt
d dt
dy dt dα dt
=
+α x
+α x
+L+α x
.
α
y
x
x
x
1
k
2
1
α , from the above expression we have
Gα = G − α G − α G − L − α G
Where,
Gα
1
1
2
k
2
α
is the growth rate of
the growth rate of output
the output elasticity of the
y; G
i
th
k
2
Because it is difficult to measure
y
k
2
1
i
( 41)
k
or the growth rate of technology;
is the growth rate of the
factor, i.e.,
i
th
factor;
G
α
i
y
is
is
x ∂y , i = 1, 2,L, k .
=
α
y ∂x
i
i
i
Because the parameter estimation of the Cobb-Douglas production function is
not the emphasis of this paper, we do not discuss it further any more.
24
5. Conclusions
Based on the Cobb-Douglas production function we can get a series of such
cost functions as the long-run total cost, the long-run average cost, the long-run
marginal cost, the short-run total cost, the short-run average cost, and the
short-run marginal cost. Based on these cost functions we can accordingly get
cost curve graphs, the long-run equilibrium input quantity employed and the
short-run equilibrium input quantity employed. However, all of these
presuppose the parameter estimation. Different parameters
different
B
k
and
B
j
, whereas different
B
k
and
B
j
α
i
result in
result in different
long-run and short-run cost curve graphs. In addition, compared to the
estimation of the parameters
α , the estimation of the parameter α
i
has its
own special natures.
25
6. References
BAIRAM, ERKIN I. and McRAE, SHAUN D., (1999), “Testing the
convergence hypothesis: a new approach”, Economics Letters, 64, 351-355.
Bourguignon, Francois., (1974), “A Particular Class of Continuous-Time
Stochastic Growth Models”, Journal of Economic Theory, 9, 141-158.
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