Warning: this is very much not my area. I don't do micro/GE theory. I was reading a bit of this stuff over 30 years ago, and then moved on to other things. If somebody asked me: "So Nick, what was all that about?", this is what I would say. Take it in the spirit of a personal reflection/interpretation. It still puzzles me.
1. Some economists in Cambridge UK wanted to explain prices without talking about preferences. I don't know why they didn't want to talk about preferences.
2. They made some special assumptions that helped them explain prices from technology alone, without talking about preferences. Like: all labour is identical; all technology is linear; prices never change over time.
3. But they still couldn't explain the rate of interest. Because it's hard to explain the rate of interest if you don't want to talk about time preferences. And all the other prices depend on the rate of interest, as well as on technology. So they assumed the rate of interest was exogenous.
4. Some economists in Cambridge US made a very special assumption that let them explain the rate of interest without talking about time preferences. They assumed that there was only one good, and it could be converted back and forth between the consumption good and the capital good by waving a wand. This very special assumption meant that the price of the capital good was always the same as the price of the consumption good, and that the rate of interest was determined by the marginal product of capital.
5. You might have thought that this would make the economists in Cambridge UK happy. Because they could now explain the rate of interest without talking about preferences. But they were unhappy.
6. A lengthy debate followed. [This is a good line to remember if you are ever asked to take the minutes at a departmental meeting.]
7. Eventually it was agreed that the economists at Cambridge US had made a very special assumption. And that what they said about what determined the rate of interest wouldn't be true without some special assumption like that.
8. Everyone went back to doing what they were doing before the debate took place. Everyone forgot about the debate, and nobody could remember what it was about. Except the economists at Cambridge UK, who felt that they had won.
9. It doesn't make any sense to me either.
10. I think it was maybe about politics. The rate of interest is a touchy subject, politically.
11. I think that if you want to explain prices, including interest rates, then you really need to talk about preferences, including time-preferences, as well as technology. Unless you are willing to make some very special assumptions about technology.
A question: do economics departments actually teach capital theory nowadays? I don't mean one-good models like the Solow or AK Growth models; and I don't mean existence proofs of Arrow-Debreu General Equilibrium. I mean something in between those two extremes. I mean something vaguely like Dutch Capital Theory. What do they teach?