Prompted by this exchange where /u/chaosmosis and /u/louisanderson make their concerns about employment and automation known.

As /u/besttrousers suggests we'll use the Cobb Douglas production function and a logarithmic utility function. We'll have two products (Apples and Oranges) along with two producers (Humans and Robots). The production functions for humans producing apples will look like this:

[; Y_{HA}:= A_{HA} L_{HA}^{\beta_{HA}} K_{HA}^{1-\beta_{HA}} ;]

Along with a utility function of:

[; U(L,K)=U_{A} log(Y_{HA}+Y_{RA})+U_{O} log(Y_{HO}+Y_{RO}) ;]

Now, before we start doing any math lets address the concerns the commentators have with the model.

There can be so many robots

I think you might be assuming there is a finite AI sector with limited capacity.

This is trivially true, resources are finite so AI is finite.

I'm not saying there's a fixed amount of work, I'm saying there's a practically unbounded ability to substitute mechanical or AI for human labor.

The commentator here acknowledges that jobs are bounded only by human wants while also maintaining that practically all these jobs can be done by robot labor. Here he is assuming literal post-scarcity of labor in which case economics is over.

Robots are cheaper better faster stronger

But we are explicitly talking about a scenario where AI is cheaper than maintaining human labor.

Another concern is that the model doesn't address the fact that robot labor can be better and cheaper than human labor. When it comes to expense this isn't exactly the case as $1 of human labor is exactly as expensive as $1 of robot labor. Robot labor however can be more productive per $1. This is addressed by the production multiplier (A) of each respective production function.

The math

Lets start by defining humans not having jobs. We'll say that humans will not have jobs if no capital is allocated to humans producing either apples or oranges.

While we could construct the bounds and find the maximal points of the utility function we're not going to go through all that. We're just going to show that the optimal distribution of capital is such that some is distributed to human production. Lets start by taking a look at two partial derivatives:

[; \frac{dU}{dK_{HA}}=\frac{U_{A}}{Y_{HA}+Y_{RA}} A_{HA} L_{HA}^{\beta_{HA}}(1-\beta_{HA})K_{HA}^{-\beta_{HA}} ;]

[; \frac{dU}{dK_{RA}}=\frac{U_{A}}{Y_{HA}+Y_{RA}} A_{RA} L_{RA}^{\beta_{RA}}(1-\beta_{RA})K_{RA}^{-\beta_{RA}} ;]

This gives us the result that

[; \frac{dU}{dK_{HA}} > \frac{dU}{dK_{RA}} \iff \frac{A_{HA}L_{HA}^{\beta_{HA}}(1-\beta_{HA})}{A_{RA}L_{RA}^{\beta_{RA}}(1-\beta_{RA})} > \frac{K_{HA}^{\beta_{HA}}}{K_{RA}^{\beta_{RA}}} ;]

Since, by symmetry, we can assume that LHA is not 0, then we can see that for sufficiently small KHA it is more efficient to move capital from robots producing apples to humans producing apples.

Shortly on comparative advantage

Here a user talks about why comparative advantage doesn't apply to this case but I don't really see where he's going with it. As you can see there is no need to consider comparative advantage for it to be efficient to employ humans at all. Comparative advantage does however explain which jobs humans will choose in the maximization of the utility function.

If anyone is interested the exact maximal value of the utility function should be pretty easy, if a little time consuming, to find using Lagrange multipliers along with the obvious bounds on capital and labor.