What is α in a Cobb-Douglas utility function?

Question

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

Answer 1

For Cobb-Douglass $\alpha$ can be shown to give you the share of budget devoted to consumption of one of the good since setting $z_1=x_1-k$ and $z_2=x_2-k$ and using budget constraint $\sum p_i x_i=m$ we can easily see that by solving this constrained problem by Lagrangian we get optimal demands given by:

$$z_1^*=\alpha \frac{m}{p_1} \implies \alpha = z_1^* \frac{p_1}{m}$$

So this tells you what fraction of income is devoted to consuming $z_1$ which is consumption of the good $x_1$ minus the factor $k$. Equivalently for $z_2$ we will have $1- \alpha = z_2^* \frac{p_2}{m}$.

As regards to $k$ that is some factor that represents the minimum amount of $x_1$ and $x_2$ that has to be consumed for it to give you positive utility. It’s definitely not capital as that would not make sense, rather it is something like being required to consume some minimum amount of a medicine for it work or minimum amount of food to start feeling positive utility from consumption or something along those lines. What exactly k is should be mentioned in the paper where you saw it.