Chap. 12 Differentiation and total differentiation · total differentiation. 2 Total...

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1 Chap. 12 Differentiation and total differentiation dx x f dy x f y ) ( ) ( ) , ( y x f z dy y z dx x z dy f dx f dz y x 2 2 1 1 2 1 ) , ( dx f dx f dy x x f y total differentiation

Transcript of Chap. 12 Differentiation and total differentiation · total differentiation. 2 Total...

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Chap. 12 Differentiation and

total differentiation

dxxfdyxfy )()(

),( yxfz

dyy

zdx

x

zdyfdxfdz yx

221121 ),( dxfdxfdyxxfy

total differentiation

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Total differentiation

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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

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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)

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|}||,min{|

),(

yx

yxfz

Not totally differentiable at (0,0)

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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

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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)

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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

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The 2 element investing product

function

),( LKFQ

output :Q capital ofinput :K

labor ofinput :L

LKQ 2

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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 ,

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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

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CES type production function

CES type production function

)(),( LKALKF 0,,,, A

Marginal productivity of capital

Marginal productivity of labor

11)(),( KLKALKFK

11)(),( LLKALKFL

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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

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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

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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

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Homogeneous function

Function y=f(x1,x2) is n-order homogeneous

),(0),(),( 212121 xxxxfxxf n

LKLKFQ 2),(Given

),(22),( LKFLKLKLKF

1st order homogeneous

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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

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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

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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

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Special examples of CES

production function①

linear product function

② and

Cobb-Douglas type production function

1

1 0

LAKALKF ),(

LAKLKF ),(

bLaK

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③ and

Leontiev type production function

},min{ ),( LKALKF

1

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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?

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Ka

K0

00 Ka

aL

K

L

Q

L0

24

L

0 K

Q=Q0

Q=Q1

Q0<Q1

K0

L0

aL/aK

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x1

x2

yy=f(x1,x2)

iso-quant curve

y0

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x2

0 x1

y=y0

y=y1

y0<y1

dy=f1(x10,x2

0)dx1+f2(x10,x2

0)dx2

x10

x20

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x1

x2

y

x10

y=f(x10,x2)

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y

0 x2

x10<x1

1

y=f(x11,x2)

y=f(x10,x2)

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x1

x2

y

y=f(x1,x20)

x20

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y

0 x1

x20<x2

1

y=f(x1,x21

)

y=f(x1,x20)

y=f1(x10,x2

0)

x10