How does economic theory handle full automation of production resulting in full unemployment? As advances in robotics and artificial intelligence continue, one possible outcome is a laborless economy, in which capital (means of production) is the dominant factor in production, by multiple orders of magnitude.  Looking at the current system, it would seem to me that a steady progression toward full automation within the bounds of the current market structure would lead toward an ever increasing concentration of wealth amongst those that possess capital.  (coincidentally, this seems to be occurring)  The end result of this, in my mind, seems to be a collapse of capitalism, in which supply curves are vertical, demand curves are horizontal, and they intersect near 0.

Now there are proposed solutions, which seem to move toward socialism, in general.  In this question, I am not seeking one of these solutions, but rather, theory.

My question is, it seems that traditional economic theory relies on { capital, labor }.  How does economic theory handle a zero-labor scenario, in which the capital is essentially a black box that outputs products?
If this were to occur, there are three things we need to look at.

1) Will production occur?

2) Without wages, how will households make money?

3) What will be the welfare effects?

$\textbf{1)}$ To answer the first question, if we assume that capital replaced all labor, we are assuming that capital and labor are substitutes which means that the production function will have the following form:
$$F(k,l)=\alpha k+\beta l$$
What this means is that if $l=0$, $F(k,l)$ will still be greater than zero, so production will still occur.

$\textbf{2)}$ To answer the second question, we can look at how households make money. Households make money from wages, and they make money from owning a fraction of firms (so they get that fraction of the profits the firm makes). All firms are owned by households (either through stocks or private ownership). The typical income of a household looks like the following:
$$Y(\frac{}{})=wl+\theta\Pi$$
Where $w$ is wages, $l$ is labor hours, $\theta$ is the fraction of firms which the household owns, and $Pi$ is those firms' profits. If $l=0$, our income reduces to $Y(\frac{}{})=\theta\Pi$

One thing that immediately comes to mind is "without wages, won't all households' income decrease?" Well, we don't know for sure. According to economic theory, the reason the automation will have occurred is because it will cost less. What this means is that profit will increase, so $\theta\Pi$ will also increase. It really depends on whether $\Delta\theta\Pi$ is larger than, smaller than, or equal to $wl$.

$\textbf{3)}$ The welfare effects are where we run into problems. When we look at the equation for income of a household, we can conduct a thought experiment: How does the fraction of income attributed to each term change when in different income groups? In the highest income groups, most of their income will come from $\theta\Pi$. An example of this would be the show SharkTank. In this show, companies come to the very wealthy "sharks" looking for capital, and in turn, they often offer a share of their company. Now, when we think about lower income groups, where does most of their income come from? Well, most of their income comes from wages because they cannot afford to save money (and therefore invest in stocks or entrepreneurship). What would happen as a result from full automation, we would theoretically see the income inequality grow to be even larger than it currently is. This is a very undesirable result.