(Sorry about the title. The devil made me write it.)
What are we afraid of? Let's think about the worst-case, nightmare scenario for the distribution of income.
Assume that all capital is robots, and robots are perfect substitutes for human workers. One robot can produce everything and anything one human worker can produce. And that includes producing more robots.
And assume that every year the technology of robot production improves, so that it takes less and less time for one robot to produce another robot.
That sounds nightmarish, right? Because robots will get cheaper and cheaper, and drive down human wages?
Well, no. They won't. Or rather, it all depends. It depends on whether we add other forms of capital, or land, to the model.
Labour and Robots only.
Let's start out by ignoring land. And the only form of capital is robots. You can produce everything with just human or robot labour.
The production function is: C + I/a = L + K and Kdot = I
where C is consumer goods produced per year, I is robots produced per year, a is a parameter which increases over time as technology improves and robots get easier to produce, K is the stock of robots, and L is the number of human workers.
There's another way to look at the parameter a. It's the rate at which robots can reproduce themselves if they aren't producing consumption goods instead. (I'm assuming that robots can't, er, reproduce and make chewing gum at the same time.)
Let's measure wages in terms of consumption goods. Because consumption is what people care about. Robots and humans earn the same wages. Since both robots and human workers produce one unit of consumption goods per year (or per day, or per hour, or whatever) their Marginal Products and wages will be one unit of consumption goods too.
W = 1
In this simple model, improving technology for producing robots has no effect whatsoever on wages.
Not at all nightmarish, is it?
It will however have an effect on the rate of interest.
We know that the price of a robot in terms of consumption goods will be 1/a. (That's because I have assumed a linear PPF between consumption and robots, so the opportunity cost of producing one extra robot is always (1/a) units of consumption).
Suppose a is rising over time, so (1/a) is falling at rate g. We know that each robot earns 1 unit of wages per year. So the rate of interest (measured in terms of consumption goods) Rc must equal the rate of return on owning a robot, which is annual robot wages (1), divided by the price of a robot (1/a), minus the rate of capital losses from the falling price of robots, so:
Rc = a - g.
In this simple model, the rate of interest is determined by the rate at which robots reproduce, and by the rate of change of the rate at which robots reproduce. The bigger is a (the quicker robots reproduce) the higher the rate of interest. The faster a is rising (the quicker the rate of technological change in robot reproduction) the lower the rate of interest. If g is positive but constant, the rate of interest will be rising over time.
It's simpler if we measure interest rates in terms of robots, Rr, because then we can ignore the fact that the price of robots will be falling over time. Since one robot can produce a robots per year,
Rr = a
The interest rate, measured in terms of robots, will be rising over time if technological change increases the rate at which robots reproduce.
Labour and Robots plus other Capital.
Robots are a form of capital goods that are perfect substitutes for labour. What happens if we introduce a different form of capital that is a complement to labour?
The simplest way to do this is to assume there is a one-year lag between humans and robots doing the work and the extra consumption and new robots being produced. So the production function now becomes:
C(t) + I(t)/a = L(t-1) + K(t-1)
The wage, measured in terms of current consumption, now becomes the present value of the (future) Marginal Product of Labour:
W = 1/(!+Rc)
The rate of interest Rc must equal the rate of return on owning a robot, which is the wage of a robot (W) divided by the price of a robot (1/a), minus the capital losses from the falling price of robots, g:
Rc = a/(1+Rc) - g
I think (somebody please check my math) that Rc, as before, is increasing in a and decreasing in g. That means that if a is growing at a constant rate, the rate of interest will be rising over time.
And, since W=1/(1+Rc), that means that wages (in terms of consumption) will be falling over time.
OK. That's a much more nightmarish scenario. For those who only own their own labour.
But it's not very realistic, for recent years, because real interest rates (deflated by the CPI) have not been rising. They have been falling.
Labour and Robots plus Land.
OK, let's scrap the lag in the production function, but put land (Natural Resources, N), along with labour plus robots, into a Cobb-Douglas production function:
C + I/a = (L + K)b.N1-b
It's a constant returns to scale production function, but holding land fixed we get diminishing marginal returns to labour plus robots. (I have implicitly assumed, by making the PPF between C and I linear, that producing consumption goods and robots are equally land-intensive.)
The human (or robot) now earns a wage equal to the Marginal Product of Labour:
W = b(N/(K+L))1-b [edited to fix math error spotted by Kathleen.]
As the number of robots increases, the wage gets driven down by diminishing returns, just as in Malthus/Ricardo, except it is the robot population that is increasing over time, if people save and invest in building more robots.
With a little bit of math, we can show that human plus robot workers earn a constant share b of total output, and landlords earn the remaining constant share (1-b). But as more robots are built, and the robot/human ratio K/L rises, human workers earn a decreasing share of b. And as total output expands, land rents per acre rise.
And the rate of interest is:
Rc = ab(N/(K+L))1-b - g [edited to fix math error]
To figure out whether Rc is rising or falling over time we need to figure out if the growing stock of robots is making the denominator grow more or less quickly than the numerator of the first term. And that will depend on people's consumption/savings choice, which in turn depends on their intertemporal consumption preferences. The math is beyond me, but I'm pretty sure the effect could go either way. (To figure it out, we need an additional equation representing intertemporal preferences in which Rc is an increasing function of the growth rate of consumption.)
Anyway, if you are looking for a nightmare scenario that is at least vaguely realistic, robots alone won't do it. I think you need to go back to Malthus/Ricardo, and put land back into the model.
That's what I was trying to say way back in this old post. I've just said the same thing with more math.
(I don't do micro, dammit, or growth theory (which is really micro, despite what the macro textbooks say). Why am I doing micro?)