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Chap. 12 Differentiation and
total differentiation
dxxfdyxfy )()(
),( yxfz
dyy
zdx
x
zdyfdxfdz yx
221121 ),( dxfdxfdyxxfy
total differentiation
3
Total differentiation and the
tangent plane
)0,0(),( ),( yxyxfz
dyfdxfdz yx )0,0()0,0(
)1),0,0(),0,0(( toorthogonal is ),,( yx ffdzdydx
),,(by expanded plane the
vector tonormal a is )1),0,0(),0,0((
dzdydx
ff yx
4
Partial differentiability and total
differentiability
|}||,min{|),( yxyxfz
0),0(,0)0,( yfxf
0)0,0(,0)0,0( yx ff
partially differentiable at (0,0)
6
Continuous differentiability
Partial derivatives f1(x1,x2) , f2(x1,x2) of f(x1,x2) are continuous
⇔ f(x1,x2) is continuously differentiable
(C1 class).
f(x1,x2) is continuously differentiable
⇒ f(x1,x2) is totally differentiable
7
High order (n times) continuous
differentiability2nd partial derivatives f11 , f12 , f21 , f22 of f(x1,x2) are continuous⇔ f(x1,x2) is twice continuously differentiable
f(x1,x2) is twice continuously differentiable
⇒ f12 =f21All n partial derivatives of f(x1,x2) are continuous
⇔ f(x1,x2) is n times continuously differentiable
f(x1,x2) is n times continuously differentiable
⇒ fij =fji (Young’s theorem)
8
Differentiation of the composite
function of 2 variable functions
)()(),( tyytxxyxfz
function s' )())(),(( ttFtytxfz
dt
dyf
dt
dxf
dt
dztF yx )(
dt
dy
y
f
dt
dx
x
f
The chain rule
9
The 2 element investing product
function
),( LKFQ
output :Q capital ofinput :K
labor ofinput :L
LKQ 2
10
Marginal productivity
K
LKFLKFMP KK
),(),(
Marginal productivity of capital MPK
Marginal productivity of labor MPL
L
LKFLKFMP LL
),(),(
,2For LKQ
L
KMP
K
LMP LK ,
11
Cobb-Douglas type production
function
LAKLKF ),(
Cobb-Douglas type production function
0,, A
Marginal productivity of capital
LAKLKFK
1),(
Marginal productivity of labor
1),( LAKLKFL
12
CES type production function
CES type production function
)(),( LKALKF 0,,,, A
Marginal productivity of capital
Marginal productivity of labor
11)(),( KLKALKFK
11)(),( LLKALKFL
13
Question 1
LAKLKF ),(
Seek the condition that the law of diminishing
maiginal product.
Partial derivative of the marginal productivity
of capital w.r.t. capital LAKLKFKK
2)1(),(
0)1(0),( LKFKK
Cobb-Douglas type production function
0,, A
10
14
Question 2
CES type production function )(),( LKALKF 0,,,, A
))1()1(()(
),(
22 LKKLKA
LKFKK
0),( LKFKKfor any K, L
)1,1(or )1,1(
Partial derivative of the marginal productivity
of capital w.r.t. capital
15
The law of return to scale
1),(),( LKFLKF
1),(),( LKFLKF
1),(),( LKFLKF
F(K,L) is increasing return to scale
F(K,L) is constant return to scale
F(K,L) is decreasing return to scale
16
Homogeneous function
Function y=f(x1,x2) is n-order homogeneous
),(0),(),( 212121 xxxxfxxf n
LKLKFQ 2),(Given
),(22),( LKFLKLKLKF
1st order homogeneous
17
The law of return to scale and
homogeneous production functions
1n
If Q=F(K,L) is n homogeneous and n>1
1n
1n
1),(),(),( LKFLKFLKF n
increasing return to scale
constant return to scale
decreasing return to scale
18
The law of return to scale and Cobb-
Douglas type production function
LAKLKF ),(
Cobb-Douglas type production function
0,, A
),()()(),( LKFLKALKF
1 1 1
shomogeneouorder )(
increasing return to scale
constant return to scale
decreasing return to scale
19
The law of return to scale and CES
type production functionCES type production function
)(),( LKALKF 0,,,, A
))()((),( LKALKF )()( LKALKA
shomogeneouorder
111
),( LKF
increasing return to scale
constant return to scale
decreasing return to scale
20
Special examples of CES
production function①
linear product function
② and
Cobb-Douglas type production function
1
1 0
LAKALKF ),(
LAKLKF ),(
bLaK
22
Appendix: Leontiev type
production function
LKL
LKK
LKK
LKaLaKifaL
aLaKifaK
aLaKifaK
a
L
a
KQ
,min
Q
La
Q
Ka
a
L
a
KQ LK
LK
,
non-differentiable
product function keeping capital constant at K0?