Chap. 12 Differentiation and total differentiation · total differentiation. 2 Total differentiation. 3 Total differentiation and the tangent plane ... Partial derivative of the marginal

1

Chap. 12 Differentiation and

total differentiation

dxxfdyxfy )()(

),( yxfz

dyy

zdx

x

zdyfdxfdz yx

221121 ),( dxfdxfdyxxfy

total differentiation

2

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)

5

|}||,min{|

),(

yx

yxfz

Not totally 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

21

③ and

Leontiev type production function

},min{ ),( LKALKF

1

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?

23

Ka

K0

00 Ka

aL

K

L

Q

L0

24

L

0 K

Q=Q0

Q=Q1

Q0<Q1

K0

L0

aL/aK

25

x1

x2

yy=f(x1,x2)

iso-quant curve

y0

26

x2

0 x1

y=y0

y=y1

y0<y1

dy=f1(x10,x2

0)dx1+f2(x10,x2

0)dx2

x10

x20

27

x1

x2

y

x10

y=f(x10,x2)

28

y

0 x2

x10<x1

1

y=f(x11,x2)

y=f(x10,x2)

29

x1

x2

y

y=f(x1,x20)

x20

30

y

0 x1

x20<x2

1

y=f(x1,x21

)

y=f(x1,x20)

y=f1(x10,x2

0)

x10