Translog Production Function

7/25/2019 Translog Production Function

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1

Derivation of Translog form from CES form

The constant elasticity of substitution form of production function can be written as:

1

1 1 2 2y A X X

= + .(B1)

The CE form shown abo!e has constant elasticity of substitution "i!en by#

1

1

=

Ta$in" the lo" on both sides of the e%uation (B1) it becomes#

1 1 2 2

1ln ln ln( )y A X X

= + .(B2)

&et 1 1 2 2( ) ln( )f X X = + ...(B')

1 1 1 2 2 2

1 1 2 2

ln ln( )

X X X X f

X X

=

+

..(B)

( ) ( )

( )

2

1 1 2 2 1 1 1 1 1 1 2 2 2 1 1 1 2 2 2

2

1 1 2 2

( ) (ln ) ln ln ln ln( )

X X X X X X X X X X X Xf

X X

+ + + =+

E*pandin" +f, in a Taylor series around - and ta$in" terms upto the second order:

Taylor series around - upto second order can be "i!en as:

2(-) (-)

( ) (-) 1. 2.

f f

f f = + + .. (B/)

1 2(-) ln( )f = +

1 2 2

1 2

ln ln(-)

X Xf

=

+

( )( ) ( )

2 21 21 2 1 22

1 2

(-) ln ln 2 ln lnf X X X X

= + +


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2

( )

22 21 2 2 1 2

1 2 1 2 1 221 2 1 2

ln ln( ) ln( ) (ln ) (ln ) 2 ln ln

2

X Xf X X X X

= + + + + + +

ubstitutin" the !alue of ( )f in e%uation (B2)#

( )( )

( )( )

( ) ( )

21 2 1 2 1 21 2 12

1 2 1 2 1 2

21 2 1 2 1 22 1 2 1 22 2 2

1 2 1 2 1 2

ln( ) 1ln ln ln ln ln

2

1 1 1ln ln ln ln ln

2 2 2

y A X X X

X X X X X

+= + +

+ + +

+ ++ + +

( ) ( ) ( ) ( )2 2-- -1 1 -2 2 11 1 22 2 12 1 2 21 1 2ln ln ln ln ln ln ln ln lny a a X a X a X a X a X X a X X= + + + + + +

0rom the e%uation some restrictions should be applied#

( ) ( )

( )

( )1 21 2

-1 -2

1 2 1 2 1 2

1a a

++ = + = =

+ + +

( ) ( )1 2 1 2

11 12 22 212 2

1 2 1 2

-a a a a

+ = + = + =

+ +

( )1 2

12 21 2

1 2

a a

= =

+

These are the conditions of homo"eneity of de"ree one in prices and they are necessary

to maintain symmetry of cross effects.

Testing of Stolper Samuelson Theorem using Translog Production

Function Form:

&et there be "oods and factors of production. The translo" function can be written

as:


3/8

3

= = = =

== = =

+

++++=

M

k

M

l

N

i

M

k

kiiklkkl

M

k

kkj

N

i

N

i

N

j

iijii

VpcVVb

VbppapaaG

1 1 1 1

1

-

1 1 1

---

lnlnlnln2

1

lnlnln

2

1lnln

3here iP is price ofthi "ood and kV is the supply of

thk factor. ymmetry of cross

effects re%uires:

jiij aa =

4nd lkkl bb =

&inear homo"eneity in 5 and 6 re%uires#

==i k koi ba 1-

======i j k l i k ikikklklijij ccbbaa -

4ssumin" output of "ood i :

( )ii Vfy =

7utput is the function of factor inputs. 8ere ob9ecti!e is to ma*imise the ;6 function#

=

=N

i

iiypG1

ub9ect to: VVi # where 5 is the total factor input. is ma*imum when ine%uality

holds#


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4

( )

==

N

i

ii VfpG1

( ) ( )=

+=N

i

iii VfpVfp2

1

( )==

+

=

N

i

ii

N

i

i VfpVVfp22

1

0rom the en!elope theorem# we can differentiate this with respect to p and V # holdin"

the optimum input choices iV constant.

i

i

yp

G=

iiii

ii

sGpy

Gp

pG

pG === lnln

This is share of "ood i in ;6.

imilarly# kk

wV

G=

This is price of ith factor.

kkkk

ki

sG

Vw

G

V

V

G

V

G==

=

ln

ln

This is share of ;6 de!oted to factor $.

Derivation of Share Equation:


5/8

5

==

++=

=

M

k

ik

N

j

jiji

i

i Vcpaa

p

Gs

11

- lnln

ln

ln


6/8

6

==++++=

N

i

kiik

M

l

lklkkk pcpcVbVabs2

11

2

11- lnlnlnln

= == =

++=N

i

N

i

iikik

M

l

M

l

klklkk pcpcVbVbbs2 2

1

2 2

11- lnlnlnln

= =

+

+=

M

l

N

i

i

ik

l

klikp

pc

V

Vbbs

2 2 11

- lnln

There are =1 independent e%uations.

Derivation of Stolper Samuelson Elasticities:

3a"es can be e*pressed as:

k

k

k V

Gs

w =

The tolper amuelson elasticities can be e*pressed as#

k

i

i

k

i

kki

w

p

p

w

p

we

=

=

ln

ln

i

k

kk

i

pGs

Vwp

+

=

i

k

i

k

k

i

p

sG

p

Gs

Gs

p

0rom e%uation .. we ha!e ( )GsVw kkk =


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7

i

k

k

i

i

iki

p

s

s

p

p

G

G

pe

+

=

i

ii

i

ki sp

G

p

G

G

pe =

=

=ln

ln

++

=

= =

M

l

M

i

iik

i

klok

ii

k

p

pc

V

Vbb

pp

s

2 2 1

1 lnln

;ifferentiatin" e%uation with respect to ip

k

ik

i

ik

k

i

i

k

k

i

s

c

p

c

s

p

p

s

s

p==

ubstitutin" e%uation and in we "et

k

ik

ikis

cse +=


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Documents

Translog Production Function