The price of basic necessities is limited by the cost of production, not some fundamental scarcity. We could produce as much food, clothing etc. as anyone could want without increasing the cost. (In fact, the cost should decrease slightly due to economies of scale.)

If UBI truly is basic income that is livable then labor costs will go up, and thus production costs go up.

It's not that simple though. On the one hand, UBI will make people less willing to accept work, but on the other hand it will also make them able to have a higher quality of life with a lower wage.

If someone is currently on $25k and gets a $10k basic income, their employer can lower their salary to $20k and they'll still be getting $5k more than they were before. (This won't happen immediately because people will be reluctant to take pay cuts, but I think it will in the longer term.)

Exactly how much of this difference the employee and employer capture in the longer term is a concern, but I think giving people the option to quit without starving will go a long way towards empowering them to insist on fair pay. I also think a result would be an increase job quality, because I think a lot of people would take a pay cut to work somewhere less awful if they could afford it.

Of course that money comes from taxes, so there's still a cost somewhere, but I'm not as convinced progressive taxes on the wealthy will have quite as big an impact on the costs to produce basic goods.

The price of basic necessities is based on what people will pay for it (or what merchants can charge for it). If everyone has $10 today to buy groceries and all of a sudden, tomorrow everyone has $100 to buy groceries, what do you think is going to happen to the price of groceries?

Nothing much in the long run, competition between supermarkets is pretty cutthroat and it should level out not too far from where it is now.

If UBI is paid for with inflation than prices will go up to capture an equivalent in value. If UBI is paid for by increased taxes then all businesses will need to raise prices.

price = f(cost) is incorrect cost = f(price) is correct

This is basic economics. (Seriously, this is in the easy stuff like Hazlitt.)

For an example, see joe's response.

Just to check that I'm understanding you correctly, you're saying that the cost of producing a product is a function of its sell price?

Yes. Costs are determined by the sale price, not the other way around. For example, if the sale price of a product increases, the demand for its components raises, which raises their cost.

If a would-be producer cannot source materials that would cost him less than the sale price, he just won't build the product.

[1] has more details on the subject.

[1] https://mises.org/library/whats-cost-got-do-it-0