cobb-douglas production function 1 suppose that the production function is q = f(k, l ) = ak a l b...
TRANSCRIPT
Cobb-Douglas Production Function
1
Suppose that the production function is
q = f(k,l) = Akalb A,a,b > 0This 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 scaleif a + b > 1 increasing returns to scaleif a + b < 1 decreasing returns to scale
Cobb-Douglas Production Function
2
The Cobb-Douglas production function is linear in logarithms
ln q = ln A + a ln k + b ln la is the elasticity of output with respect to kb is the elasticity of output with respect to l
CES Production Function
3
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
4
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,
5.0
5.0
)/(1
)/(1
l
ll
k
k
f
fRTS
k
Technical Progress
5
Methods of production change over timeFollowing the development of superior
production techniques, the same level of output can be produced with fewer inputsthe isoquant shifts in
Technical Progress
6
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 lchanges in A over time represent technical
progressA is shown as a function of time (t)dA/dt > 0
Technical Progress
7
Differentiating the production function with respect to time we get
dt
kdfAkf
dt
dA
dt
dq ),(),(
ll
dt
df
dt
dk
k
f
kf
q
A
q
dt
dA
dt
dq l
ll),(
Technical Progress
8
Dividing by q gives us
dt
d
kf
f
dt
dk
kf
kf
A
dtdA
q
dtdq l
l
l
l
),(
/
),(
///
l
l
l
l
ll
dtd
kf
f
k
dtdk
kf
k
k
f
A
dtdA
q
dtdq /
),(
/
),(
//
Technical Progress
9
For any variable x, [(dx/dt)/x] is the proportional growth rate in xdenote this by Gx
Then, we can write the equation in terms of growth rates
ll
l
llG
kf
fG
kf
k
k
fGG kAq
),(),(
Technical Progress
10
Since
llGeGeGG qkkqAq ,,
kqeq
k
k
q
kf
k
k
f,),(
l
l
l
ll
l
l ,),( qeq
q
kf
f
Technical Progress in the Cobb-Douglas Function
11
Suppose that the production function is
q = A(t)f(k,l) = A(t)k l 1-
If we assume that technical progress occurs at a constant exponential () then
A(t) = Ae-t
q = Ae-tk l 1-
Technical Progress in the Cobb-Douglas Function
12
Taking logarithms and differentiating with respect to t gives the growth equation
qGq
tq
t
q
q
q
t
q
/lnln
Technical Progress in the Cobb-Douglas Function
13
l
l
l
GGtt
kt
ktAG
k
q
)1(ln
)1(ln
)ln)1(ln(ln
Important Points to Note:
14
If all but one of the inputs are held constant, a relationship between the single variable input and output can be derivedthe marginal physical productivity is the
change in output resulting from a one-unit increase in the use of the inputassumed to decline as use of the input increases
Important Points to Note:
15
The entire production function can be illustrated by an isoquant mapthe slope of an isoquant is the marginal rate
of technical substitution (RTS)it shows how one input can be substituted for
another while holding output constantit is the ratio of the marginal physical productivities
of the two inputs
Important Points to Note:
16
Isoquants are usually assumed to be convexthey obey the assumption of a diminishing
RTSthis assumption cannot be derived exclusively from
the assumption of diminishing marginal productivityone must be concerned with the effect of changes in
one input on the marginal productivity of other inputs
Important Points to Note:
17
The returns to scale exhibited by a production function record how output responds to proportionate increases in all inputsif output increases proportionately with input
use, there are constant returns to scale
Important Points to Note:
18
The elasticity of substitution () provides a measure of how easy it is to substitute one input for another in productiona high implies nearly straight isoquantsa low implies that isoquants are nearly L-
shaped
Important Points to Note:
19
Technical progress shifts the entire production function and isoquant maptechnical improvements may arise from the
use of more productive inputs or better methods of economic organization