Question

When normalizing the parameters for a Cobb Douglas Function, prove that the utility is unique only up to a monotonic transformation if α = 0.3

Answer 1

Assume the following Cobb Douglas production function

$U= x^\alpha y^{\beta}$

The utility can be unique up to a point where $\alpha+\beta=1$

Since $\alpha$ and $\beta$ indicate the relative importance of good x and y to a consumer, the Cobb Douglas function can be written as follows.

$U=(X,Y)= x^{\Phi} y^{1-\Phi}$

Where $\Phi= {\alpha \above{2pt} \alpha+\beta}$

and $\Phi= {\beta \above{2pt} \alpha+\beta}$

Now, if $\alpha=0.3$ then $\beta=(1-0.3)=0.7$

By nomalizing $\alpha$ and $\beta$ with $\alpha=0.3$ and $\beta=0.7$, the utility will be unique upto a monotonic transformation when the Cobb Douglass production function is as follows

$$U=x^{0.3} y^{0.7}$$