Cobb.E2.80.93Douglas utilities Cobb–Douglas production function

the cobb–douglas function used utility function. in context consumer assumed have finite wealth, , utility maximization takes form:

max

x


u
(
x
)
=

max


x

i





∏

i
=
1


l



x

i



α

i






 subject constraint 



∑

i
=
1


l



p

i



x

i


=
w


{\displaystyle \max _{x}u(x)=\max _{x_{i}}\prod _{i=1}^{l}x_{i}^{\alpha _{i}}\quad {\text{ subject constraint }}\quad \sum _{i=1}^{l}p_{i}x_{i}=w}

where



w


{\displaystyle w}

total wealth of consumer ,




p

i




{\displaystyle p_{i}}

prices of goods. utility may maximized follows. first, take logarithm of utility

ln
⁡
u
(
x
)
=

∑

i
=
1


l




λ

i



ln
⁡

x

i




{\displaystyle \ln u(x)=\sum _{i=1}^{l}{\lambda _{i}}\ln x_{i}}

let λ = λ1 + ... + λl. since function



x
↦

x


1
λ





{\displaystyle x\mapsto x^{\frac {1}{\lambda }}}

strictly monotone x > 0, follows



u
(
x
)
=



u
~



(
x

)


1
λ





{\displaystyle u(x)={\tilde {u}}(x)^{\frac {1}{\lambda }}}

represents same preferences. setting




α

i


=



λ

i


λ




{\displaystyle \alpha _{i}={\frac {\lambda _{i}}{\lambda }}}

can shown that

∑

i
=
1


l



α

i


=
1


{\displaystyle \sum _{i=1}^{l}\alpha _{i}=1}

the optimal solution then:

∀
j
:


x

j


⋆


=



w

α

j




p

j




.


{\displaystyle \forall j:\qquad x_{j}^{\star }={\frac {w\alpha _{j}}{p_{j}}}.}

an interpretation of solution consumer uses fraction




α

j




{\displaystyle \alpha _{j}}

of wealth in purchasing j.

the indirect utility function can calculated substituting demand utility function. ignoring multiplicative constant depends on




α

i




{\displaystyle \alpha _{i}}

s, get:

v
(
p
,
w
)
=


w


∏

i
=
1


l



p

i



α

i









{\displaystyle v(p,w)={\frac {w}{\prod _{i=1}^{l}p_{i}^{\alpha _{i}}}}}

which special case of gorman polar form. expenditure function inverse of indirect utility function:

e
(
p
,
u
)
=

∏

i
=
1


l



p

i



α

i




u


{\displaystyle e(p,u)=\prod _{i=1}^{l}p_{i}^{\alpha _{i}}u}






^ brenes, adrián (2011). cobb-douglas utility function. 
^ cite error: named reference palgrave invoked never defined (see page).
^ varian, hal (1992). microeconomic analysis (third ed.). new york: norton. isbn 0-393-95735-7.