Cobb-Douglas Utility function
$$U=C x^a y^b$$
While applying the Cobb-Douglas formed utility function, we are actually proxy the preference of people. (The utility function is like a math representation if individuals’ preference is rational). In the utility function, we are focusing more on the Marginal Rate of Substitution between goods.
$$MRS_{x,y}=\frac{MU_x}{MU_y}=\frac{\partial U/\partial x}{\partial U/\partial y}=\frac{Cax^{a-1}y^b}{Cx^a by^{b-1}}$$
$$MRS_{x,y}=\frac{ay}{bx}$$
P.S. Cobb-Douglas gives the same MRS to CES utility function. While solving the utility maximisation problem, we take partial derivatives to the lagrangian and then solve them. Those steps are similar to calculating the MRS.
The key is that the number or value of the utility function does not matter, but the preference represented by the utility function is more important. Any positive monotonic transformation will not change the preference, such as logarithm, square root, and multiply any positive number.
Exponents Do Not Matter
The powers of the Cobb-Douglas function does not really matter as long as they are in the “correct” ratio. For example,
$$ U_1=Cx^7y^1,\quad and \quad U_2=Cx^{7/8}y^{1/8} $$
$$MRS_1=\frac{7y}{x}\quad and \quad MRS_2=\frac{7y/8}{x/8}=\frac{7y}{x}$$
Therefore, we can find that those two utility functions represent the same preference!
Or we can write \(U_1=(U_2)^8 \cdot C^{-7}\). Both taking exponent and multiplying a positive constant are positive monotonic transformations. Therefore, the powers of Cobb-Douglas do not really matter to represent the preference. (\(U=Cx^a y^{1-a}\) the exponents of the utility function does not have to be sum to one).
$$U=x^a y^b \Leftrightarrow x^{\frac{a}{a+b}}y^{\frac{b}{a+b}}$$