'The tendency of the rate of profit to fall' (TRPF) that socialists often talk about is a so-called law of economics 'discovered' by Karl Marx.
The general idea is that only human labour creates 'value' in products (the Labour Theory of Value - LTV). Marx's given reason for this in Das Kapital is pretty weak: He says that if we are able to exchange products of different types they must contain something within them that is comparable. E.g. why is 1 laptop worth the same as 2 phones? Laptops and phones are very different things, how have we been able to establish a way to compare them? Marx says it's because laptops require twice as much work to produce them as phones.
The trouble with this is that it's not hard for someone who believes that things only have as much value as someone is willing to pay for them at any given moment to come along and claim that it's demand that is comparable across products of different types, as well as the rate at which that demand can be supplied (this is the subjective theory of value - STV). The point here is that the fundamental question that the LTV deals with should not be seen as the conundrum over how we can compare and trade products of different types. That's an uninteresting conundrum and trying to solve it with the LTV doesn't get us anywhere.
I think there is a better basis for the LTV but we'll get to that below.
Another difficulty that Marx skips over is the relationship between his theory of price and his theory of value. We're told to assume that price averages out at value over time. What Marx doesn't say and what Marxists don't seem to mention very much is that the only mechanism which could make that happen would be market competition and that this law would hold more perfectly the freer the competition and less perfectly in a world of subsidies and regulations etc. In a world of subsidies and regulations it's quite possible that some things could be consistently priced at much higher than their true values and others much lower.
Anyway. Back to the TRPF.
Capitalists who own all the factories where laptops and phones get made have to pay both the wages of the workers who make the laptops and phones but also various other costs. Especially the costs of all the equipment the workers need to do their work and even for the factory itself. Marx recognised that there was a general trend whereby, over time, the proportion of capital that capitalists had to spend on wages was going down in comparison to the proportion the had to spend on all the 'stuff' (not sure if that still holds true in a world where cheap 3D printers are coming out or not?... skip over that thought for now).
The money they were having to put into the 'stuff' was only in order to remain competitive. It didn't actually produce new value in itself, only human labour working through the 'stuff' could do that. So (as the theory goes), as a smaller proportion of capital was being invested in labour a smaller proportion of their capital was actually producing them surplus value (i.e. a smaller proportion of their money was able to turn into more money through exploiting labour and not paying people equal to the full value of the things that they create).
So, back to this business about a better basis for the LTV:
There is obviously a principle involved here that hasn't been mentioned by Marx (or by any Marxists I'e ever spoken to). This principle is that there is something unique about human labour. That there is a moral factor within our economic system whereby if a person does some kind of useful productive work (making things for which there is a demand) they are owed something back for it - just by rights. Naturally we might assume that if they've lost an hour of their time (doing work that they certainly wouldn't have chosen to do for fun) then they really deserve to be paid in money that is roughly able to buy products that take an hour of someone else's time to make them (obviously we'd have to account for intensity of work and the level of skill involved, as well as the pay for any other labourers whose work was necessarily involved like people to transport stuff around etc. if we were getting serious here). So the 'value' that Marx talks about is like a debt that has been created towards the worker per 'amount' of labour they'e had to do (something that only makes sense in an economy where the worker doesn't own the things they've produced - otherwise would be no debt; you'd just want a coat and you'd make one and it would be yours). So what capitalists are really amassing is unpaid debts. If someone has to work in a bakery for an hour (the time it takes them to make 60 loaves of bread) in order to get the money to buy 1 loaf of bread the capitalist is monetising what has been held back.
It only works with humans because you can't amass debt to a machine. At the risk of sounding ridiculous; why can't you get into debt with a machine? The real reason is that machines aren't sentient. Machines just cost what they cost to run. You can't force them to do the same amount of work on half the fuel. The machine does not say to itself - I've lost an hour labouring away doing something I didn't enjoy here, I need to be compensated fairly (i.e. with the product of an hours' worth of someone else's time). Only humans who have suffered having to do the unwanted hour of work when they could have been chilling out on the beach can say that on principle they should get a fair deal. So if the human body was reduced to machine and if there were a class of people who couldn't conceive of anything more than being fed and watered and rested ready for another day of work and had no other desires at all then it would be absolutely fair to just pay them to cover this bare minimum but there is no such class of people. Everyone's life matters and everyone wants to do more than just survive. That is why something different happens when the capitalist tried to get away with paying a human with the bare minimum, it becomes exploitation.
But isn't this all getting a bit abstract? If you had a factory in which machines did everything and people did nothing (and I mean machines literally did everything: Mined raw materials, powered themselves by the sun, made the stuff and delivered the stuff straight to the consumer) would your products suddenly not have any value and not be able to be priced up and sold?
No. It's not about individual factories. If one factory did it the owner would probably be making loads of money. If every factory did it and the whole economy was fully automated (but still owned by capitalists) then market competition between them all would be driving prices down to zero eventually because there was no actual cost involved to any human person at all in the process (and this is aside from the fact that demand would completely dry up because no one would be getting any wages any more).
In a fully automated economy, eventually things would probably just have to be given away for free - or, in a darker version of this tale, the ruling class might decide that the entire working class were now just a great big waste of space and try to kill us all. Possibly with the help of their machines...
So, what have we concluded?
Various things really. Mixed bag.