[Ages ago, as a graduate student, I "wasted" my time (when I should have been finishing my thesis) reading capital theory, including the more "heterodox" stuff. My memory is pretty hazy, but I still think about the old questions occasionally. There's nothing really new here for economists familiar with "dated goods".]
1. Suppose I know about a technology that lets me convert 100 apples into 105 bananas. Should we count those extra 5 bananas as part of my income? Of course not. You can't add (or subtract) apples and oranges bananas. They are two different goods. They don't taste the same.
But if the price of one apple is one banana, then we should count those 5 extra bananas as part of my income. I can make a 5 banana profit from converting 100 apples into 105 bananas. But I can't make unlimited profits; because the faster I convert apples into bananas the higher the price of apples relative to bananas, because I am reducing the supply of apples and increasing the supply of bananas.
If a couple of dozen other people know about the same technology, I won't be able to make profits, because each of us will all use it faster and faster until the price of one apple is 1.05 bananas. I need some sort of monopoly power if I am going to earn profits from using this technology.
2. Suppose I know about a technology that lets me convert 100 apples into 105 identical apples instantly. Should we count those 5 extra apples as part of my income? Of course we should.
But I can't make unlimited profits; because the faster I use my technology to convert 100 apples into 105 apples and create 5 extra apples the lower the price of apples relative to bananas, because I am increasing the supply of apples.
If a couple of dozen other people know about the same technology, I won't be able to make profits, because each of us will all use it faster and faster until the price of one apple drops to zero bananas. Apples become so plentiful they are a free good. I need some sort of monopoly power if I am going to earn profits from using this technology.
3. Suppose I know about a technology that lets me convert 100 apples into 105 identical apples. But the technology takes one year to do its thing, so I don't get my 105 apples until next year. Should we count those 5 extra apples as part of my income?
3a. Is this like case 2, where we do count those 5 extra apples as part of my income, but I need some sort of monopoly power to make any profits?
3b. Or is this like case 1, where apples that can be consumed next year are a different good to apples that can be consumed right now, and we care about how many of each we consume, just as we care about whether we consume bananas or apples? Where one plus the (real) rate of interest is the price of apples today relative to apples next year, just like the price of apples relative to bananas?
In both case 1 and case 3, if I use my technology it's good for people who like eating bananas (apples next year) and bad for people who like eating apples (this year), because it affects the relative price of those two goods.
In case 2, if I use my technology it's good for everyone who likes eating apples (but bad for apple producers), because I am creating 5 extra apples out of thin air. And unless I have some sort of monopoly power that restricts the use of the technology, apples become so plentiful they are free.
And is "capital" a thing, or is it just the name for a process of converting apples today into apples in the future? If the 100 apples disappeared into nothingness, and reappeared next year as 105 apples, where did the "capital" go?
If apples and bananas are metaphors for apples this year and apples next year, then there is something special about technology. If we have a technology that converts 100 apples into 105 bananas, then that same technology lets us convert 100 bananas into 105 carrots, and 100 carrots into 105 dates, and so on forever. Unless we forget it.
Worthwhile Canadian Initiative 
Yes, giving up leisure for labor is exactly like giving up present consumption for future consumption. And yes, most people who give up leisure for a lot of money would probably want to give up for very little because they enjoy working, just as most people who give up present for future consumption would want to do the same because they enjoy accumulation. Neither enjoyment is modelled in mainstream models that I am aware of, although in some cases some models assume inelastic labor supply and I’ve seen some desperate pleas to put wealth into the utility function.
I agree that both arguments are obvious consequences of Arrow-Debreu but not particularly convincing as a description of what is happening in actual economies. Except that the two mirror arguments are disturbing to two mirror groups and you only chose to disturb only one. That selective argumentation is the most non-obvious and interesting part of this whole discussion.
rsj: nope. The analogue would be: if a clever person know about a technology that converts apples into twice as many bananas (or future apples) as anyone else’s technology, and keeps that technology secret, and makes profits from his monopoly power, that person would be wealthier, just like the strong worker.
Knowledge is non-rival; worker strength is rival.
Is this just saying that with perfect competition (i.e. no secrets), all forms of capital must have gross returns equal to the rate of interest, and hence on net can generate no income (to the individual capitalist)? Same as in perfect competition, no firm can make economic profit?
Nive: From that perspective, I’m asking the question: is interest income really income?
Take a model where every individual has 100 hours of labour (or land). And each individual can choose between 4 things (in any mix):
1. Consume 100 hours leisure
2. Produce and consume 100 apples today.
3. Produce and consume 105 bananas today.
4. Produce and consume 105 apples next year.
(Or any mix of the above).
Different individuals will make different choice, depending on their preferences. Why would we say that an individual who chooses 4 has higher income than an individual who chooses 2?
I would say they are all have the same income.
Is there no standard definition of income? I’m sure someone has thought about this before 🙂
I initially thought income should be defined as what you can consume, regardless of what you actually do consume. This looks like it would work nicely for 1-4 in the first period. Assuming all are price-takers, and 1 apple = 1 leisure-hour = 1.05 bananas = 1.05 apples tomorrow, they all can consume 100 apples today, so their income is 100 apples.
This does not seem to work very well in succeeding periods once you introduce savings. If #4 saves all his apples today to get 105 apples next year, and also produces 100 apples next year, we surely don’t want to say his income next year is 205 apples. Should we subtract 100, what he saved; or 105, the future value of his savings, or something else? What if he did have a secret technology, so he could produce 110 apples, is his income in the first period 110/1.05, or do we count the extra 5 apples as his second-period income?
These sort of issues look pretty similar to the issues of income accounting for a firm. What sources of income should be accrued, what should be marked to fair value etc.
From my biased point of view (working in finance), fair-value accounting together with the definition of profit & loss for a trading book would be the easiest to make consistent. For 1-3, initial wealth = 100+100/1.05. When we go from year 1 to year 2, p&l = 100 (realized) + 100 (unrealized) – (100-100/1.05) (funding) – (100+100/1.05) (yesterday) = 0. For #4, assuming he produces 100 apples next year the same as #2, in addition to the 105 coming from his investment, the breakdown is a little different, but he has the same initial wealth and p&l: 105/1.05 + 100/1.05 wealth, p&l = 105 + 100 – (205-205/1.05) – (105/1.05+100/1.05) = 0.
This does lead to the odd result that if everything is deterministic, income is always zero.
In the trading book p&l view, interest income is not really income unless it exceeds your cost of funding, by the way — that’s why we subtract funding.