Profits and income from capital are not the same thing, though they are mixed together in the national accounts. (Paul Krugman has made this point before.)
I work through some simple examples with sticky prices and/or sticky wages, where income from capital always falls in a recession caused by tight monetary policy.
[I add an incomplete postscript on how this will affect real interest rates in a recession.]
Start with a simple model.
Households own labour and capital goods. Firms pay a wage W to rent the services of labour, and pay a rental R to rent the services of capital goods, and use those services of labour and capital goods to produce consumption goods and capital goods, which they sell to households. The value of output sold, minus wages paid, minus capital rentals paid, equals firms' profits. Households own the firms, and so own those profits too.
For simplicity, assume that the supply curves of the services of the existing stocks of labour and capital goods are perfectly inelastic. Net population growth, and net investment, will make those existing stocks grow over time.
Start with full employment of both labour and capital, then assume the central bank does something stupid and there is a recession. We know that total income (equals total output) falls. But what happens to the distribution of income? That depends. On nominal rigidities.
1. Here is one extreme case. Assume output prices P are fixed, but that wages W and capital rentals R are perfectly flexible. The recession causes both W and R to fall to zero. There are unemployed workers and/or unemployed machines, both willing to work for nothing. Firms can produce output for free, but don't bother hiring the unemployed workers and machines, even at zero wages and rentals, because they can't sell the extra output.
In that extreme case, a recession causes income from wages and income from capital to fall to zero. All income is profit income. Households earn all their income from owning firms, that are able to sell some output, but not as much output as before the recession. ("But what exactly is a 'firm'?" I hear you ask. Good question.) What matters for your income in a recession is being able to sell goods, not being able to produce goods.
Now change the assumptions very slightly. Assume that firms own the capital goods they use, and pay rents to themselves. Nothing changes. We get exactly the same results as before. Except we can't tell the difference between income from profits and income from capital rentals. They both get lumped together as "profit" in the national accounts, because nobody observes the rents that firms charge themselves. But the "shadow" rental on machines -- the price that firms would be willing to pay to rent a machine -- is still zero.
2. Now let's assume that both P and W are fixed, and repeat the experiment. R is perfectly flexible. (That makes sense if firms own the capital they use, because no firm would leave its capital unemployed because it were charging itself too high a rent.)
With P and W fixed, the recession will cause unemployment of labour. It will not cause unemployment of capital, unless the technology requires labour and capital to be used in fixed proportions.
Under fixed proportions technology, capital will be unemployed, and R will drop to zero.
Under variable proportions technology, capital will be fully employed, and R will drop, but not to zero. Firms cannot sell the extra output from employing extra capital, but they can produce the same output with more capital and less labour. The cost-minimising mix of capital and labour is determined by:
MPK/MPL = R/W
(MPK and MPL are the marginal physical products of capital and labour -- the extra output per extra machine or worker employed, holding employment of the other input constant.) Since employment of labour is lower in the recession, while employment of capital in unchanged, the K/L ratio increases in a recession, so MPK/MPL falls, so R/W falls, and since W/P stays constant by assumption, R/P must fall.
3. Now let's assume that W is fixed, and that P and R are both perfectly flexible.
Under fixed proportions technology, both labour and capital will be unemployed in the recession, so R will fall to zero.
Under variable proportions technology, only labour will be unemployed, and individual firms will be able to sell as much output as they like. Profit-maximisation ensures:
MPL = W/P and MPK = R/P
Since K/L increases in a recession, MPK will fall, so R/P must fall too.
In all three cases considered above, R/P falls in a recession. Income from capital falls in a recession caused by tight monetary policy. Wage income falls in the first and second cases, and may either rise or fall in the third case. Profit income will rise in the first case, and may either rise or fall in the second and third cases (I think).
[Postscript on interest rates: Just to remind you, R is not the same as the rate of interest. It is how many dollars you pay per year to rent one machine. It is however related to the rate of interest. Let Pk be the price of capital goods, Pc be the price of consumption goods, i be the nominal rate of interest, r be the real rate of interest (deflated by Pc), and ignore depreciation, then:
i = R/Pk + Pkdot/Pk
The RHS of that equation represents the rate of return from owning a machine, which is the annual rents earned by the machine, divided by what you paid to buy the machine, plus the rate of increase of the price of machines, which will equal the nominal interest rate in equilibrium, if households are indifferent between owning machines and owning bonds. (This assumes you can actually sell your machine at the price Pk, i.e. that Pk is not sticky.)
Subtracting the inflation rate on Pc from both sides we get the real interest rate:
r = i - Pcdot/Pc = R/Pk + Pkdot/Pk - Pcdot/Pc = R/Pk + (Pk/Pc)dot/(Pk/Pc)
We only get the simple r = R/Pk = R/P = MPK in a one-good model where Pk=Pc=P, and where P is perfectly flexible.
The fall in R during a recession will tend to lower real and nominal interest rates, but the full effect on r can only be figured out if we work out what happens to Pk/Pc. In the simple one-good model, where Pk/Pc=1, we know that r will fall in a recession. In a two-good model, it will be possible for r to rise in a recession, provided Pk jumps down at the beginning of the recession, so that Pk is expected to rise faster than Pc during the recession.
Update: I think it depends on how long the recession is expected to last. If the recession is expected to last a long time, Pk/Pc will not be expected to be rising quickly, so r will fall because R falls. The IS curve slopes up. If the recession is expected to be very short, Pk/Pc will be expected to be rising quickly, so r will rise despite the fall in R. The IS curve slopes down.]