CRRA utility function with a scale parameter I am wondering if it is possible to write down a following CRRA utility function ;

$$u\left(c(t)\right)=a\frac{c\left(t\right)^{1-\sigma}}{1-\sigma}$$ where $a&gt;0$ is a constant scale parameter. I need this $a$ to have some numerical results but I am not sure if $a$ can be justified. 
In a growh model, I am trying to show the existence of Hopf bifurcation and limit cycles. Then, with a scale parameter like that, I can show that it exists. 

Normally, it does not change the usual assumptions on a CRRA utility function, (an increasing concave function)

Is there any way to justify it (are there some examples of this kind ?) or are there any types of utility function with constant scale parameters ?
As it is well-known, preferences (even under risk) are invariant to affine transformations of the utility function. Therefore, adding a 'scale' parameter adds nothing in terms of fundamental preferences (i.e. in terms of comparing streams of consumption). I cannot even imagine what kind of 'numerical results' would be sensitive to it.