What is α in a Cobb-Douglas utility function? Sorry if this is not the place to ask, I'm new here. I'm studying economy but I'm struggling to understand the Cobb-Douglas utility function.
If we've one such that xt is consumption in period t, and α and k are parameters such that 0 < α < 1 and k ≥ 0:

U(x1,x2) = α ln (x1-k) + (1-α) ln (x2-k)

What are α and k? Is α the impatience of the consumer or is it the proportion of consumption between two periods? Is k the capital or saving?
I'm using Mass Colell's book as a guide, but I can't find an explanation there and neither on the internet. If anyone could refer me to a book on this topic I'd appreciate it a lot. I only see information about the production function, not the utility one.
Thank you very much
 A: So you have a (shifted) Cobb-Douglas utility
$$U(x_1,x_2)=\alpha \ln (x_1-k)+(1-\alpha)\ln (x_2-k),\alpha\in (0,1),k\geq 0,\quad (1)$$
Note this utility function is only defined for $x_1,x_2\geq k$. Also observe utility tends to $-\infty$ as $x_1\downarrow  k$ (or $x_2\downarrow  k$). Thus, it is clear that such a consumer must consume more than $k$ units of each good, and we may interpret $k$ as the bare minimum (or infimum, I suppose) of each good that would be consumed.
Now, $\alpha$ and $1-\alpha$ may be interpreted at first glance as "preference weights" for goods 1 and 2 respectively. However, they have a much more concrete interpretation. Consider a budget constraint
$$p_1(x_1-k)+p_2(x_2-k)\leq m,$$
where $p_1,p_2$ are prices for goods 1 and 2 respectively, and $m$ is any additional income beyond the income needed to buy $k$ units of each good. Then you can show that a consumer with utility in $(1)$ would optimally choose $x_1,x_2$ satisfying $$p_1(x_1-k)/m=\alpha,\\p_2(x_2-k)/m=1-\alpha.$$
This tells us that $\alpha$ is the expenditure share that this consumer would devote to good 1 (in excess of $k$ units), and $1-\alpha$ is likewise the expenditure share this consumer would devote to good 2 (in excess of $k$ units).
