Utopian or Dystopian future?
Imagine if you will a robotic machine standing about 2m high with an adjustable height camera mast, 4 appendages, two of which operate like human hands with pressure sensitivity and delicate grip, one of which operates like a power drill/screwdriver, and one of which operates like a multi-function cutting, welding, gluing, nailing tool.
The machine has cameras on each appendage, is capable of altering its shape and height to reach anything any human standing on a stepladder or lying on a dolly could reach.
Furthermore, the machine is capable of learning many common tasks by watching people, and once one of the machines has learned a task, the "program" for that task can be copied and transferred between machines immediately. In fact, the machines have high speed wireless communication allowing them to not only transfer information between themselves, but to organize themselves into groups to carry out tasks that require cooperation. They can operate associated machinery to replicate themselves, each individual robot could assemble one new robot per day. Parts can be manufactured by teams of robots in manufacturing facilities for thousands of robots per day from relatively raw materials (steel and copper ingots, tubs full of plastic beads, recycled broken glass, each raw material produced by teams of robots working in specialized factories such as steel smelting and petroleum refineries etc etc)
Suppose furthermore that there are a variety of means put in place to avoid having these robots become dangerous dystopian Terminators. We'll suppose that some people have worked very hard to create secure protocols for controlling them, because I'm not interested in Terminator dystopia, I'm interested in a point of view about Economics.
The machine operates on batteries that take about 15 minutes to fully charge, and operates for say 4 hours between charges when doing everyday medium duty tasks. The quantity of electricity in a full charge is on the order of 20 KWh (about 17k food calories) which costs around $4 today.
Suppose the machine costs today say $100k to manufacture. It services itself (or can service another robot) for any regular servicing, and operates for 20k hrs (2.3yrs) between regular services lasting less than 1 day. Otherwise it can operate 24hrs a day with just brief recharges throughout the day.
Now, in this Utopia/Dystopia human labor is essentially meaningless. Everything from industrial welding to car washes to laundry to woodworking to clothes manufacturing to gourmet cooking to firefighting, to truck driving, to painting portraits can be done by robots using far less raw resources, far faster and more efficiently. The marginal cost of getting something done which would take one full day worth of specialized human labor would be something like $5 worth of electricity.
In a country like the US where the basic underlying assumption about the economic organization of society is that "things" are limited, and people should work to provide services to others so that they can earn money so they can buy things... If we take that point of view dogmatically to the grave with us... the Equilibrium situation would seem to be near extinction of the humans.
Why? Well, there will always be some small amount of stuff that can only be done by humans. Whether that's perhaps inventing new gourmet recipes (requires taste receptors) or validating security protocols to keep the robots from killing us all (requires humans because you can't trust the robots) there will be some small amount of work required from humans. But, for literally everything else, you could get by producing things without humans. Yet, "stuff" is still limited, like raw materials. So the equilibrium price of "stuff" is going to go to epsilon > 0 whereas the value of everyday human labor is going to go to exactly zero (you can always get the robot to do the thing cheaper).
So, sure, stuff costs next to nothing, but after a short period of time, humans have absolutely zero income and then zero financial wealth, unless they own significant quantities of natural resources and are able to charge rents for their use.
The equilibrium situation is to use the robots to compost the corpses of the everyday people who starved to death to grow crops to feed the people who happened to own land at the start of everything. And then, the economy wheels along smoothly with people who own land charging rent to the other people who own land, the only goods being exchanged are essentially very raw materials through the medium of money.
Which is to say, as the labor cost of producing stuff goes to zero, the system collapses under the assumption that people should provide valuable services in order to get the things they need to survive.
Now, suppose instead we allow each person in the world to essentially "print" a fixed amount of money in each time period. Let's just measure money in units of this fixed amount, so we can each print say exactly $1/day.
Now, in the long time equilibrium, each person can get some amount of stuff. Furthermore, the equilibrium would seem to be that the price of the stuff we produce per person would be $1. That is, ignoring the tiny number of people who actually do specialized stuff like audit the robots to make sure they don't wipe out the human race, everyone gets 1/N share of all the stuff, and as we progress, the total stuff being produced for people increases without bounds while the price of that stuff stays constant at $1 for 1/N of all the stuff produced in a day.
Whether you let people "print" money, or because it's easier to ensure that no-one is cheating if you have only one entity printing the money and distributing it, or you do some mixture of printing and taxing income, if you ensure that everyone has at least some fixed basic quantity of money coming to them every unit of time, the equilibrium of a world of plenty shifts from Dystopian nightmare collapse of the population to a tiny number (a few thousand ?) of rich people who each own 1/N of the earth's surface and exchange dollars for raw materials, to a Utopian society where everyone simply gets some of the universal plenty.
Which is to say, the Universal Basic Income (UBI) is the mathematical mechanism to eliminate a singularity in the economy at zero income. How quickly that singularity consumes our world will rest on how quickly we put people out of work. It won't take a near-magical robot. Every few years France already has rioting in the streets due to a very low labor force participation rate and high unemployment.
See some Labor Force Participation rate data here