Production of Robots by means of Robots.

(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.N^1-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?)

Comments

[K]

Nick,

Yup, this is my mental model too. Robots producing tons of stuff can never impoverish anybody. (This *ought* to be totally obvious). It's the monopolization of finite resources that is the killer. As usual, I'll refer whoever cares to Progress and Poverty. Henry George had this, and the solution, worked out over a hundred years ago.

[K]

The problem with standard neoclassical analysis is that it literally *defines* the long run to be the state in which all inputs are variable. Right off the bat, it precludes the student from thinking about natural resources. I don't know if this was a deliberate conspiracy perpetrated by the early neoclassicals on behalf of rentier interests. Georgists seem to think so. Somehow, for whatever reason, they managed to expunge land and with it the concept of rents (unearned income) from economics and the long run definition seems to be at the core of it.

[Ritwikbut]

K

"Robots producing tons of stuff can never impoverish anybody"

Not even in a model with heterogenous agents, with differential ownership of robots?

Haven't been able to spend etime on Nick's model, but think something's funny going on. Don't see how/why the total productivity of robots stays the same as the total productivity of labour across time. W isn't the issue. W*L is. But will hold back

[Nick Rowe]

K: the individual firm can vary all inputs, including land. But yes, the economy as a whole can't vary land. (So Long Run MC curves at the aggregate level will be much less elastic than at the individual firm level.)

But the lefty Sraffian model has only labour and time as inputs. If it was a conspiracy, it was a very big conspiracy. Nah, it's just townie kids, who think food comes from the supermarket, and think that rural idiocy is just a disappearing remnant of the past! Or maybe it's hard to separate out land and capital returns in National Income accounting, so Solow had to ignore land to do Solow Growth Accounting and build the Solow Growth model.

Ritwikbut:

Change the production function to C + I = L + aK

We get Wh = 1 for human wages, Wr = a for robot wages, Price of one robot = 1, and Rc = a - g as before.

[nemi]

Im not sure what you are asking

If “a” stayed constant, you would want to expand them in a way that kept:

Pk*K/(1-b) = Pr*(R+L)/b

The percentage increase in R would be bigger than that of K (but the percentage increase of R+L would be the same).

If Pr decreased over time with g, I guess that you would want:

r*Pk*MP.K=(r-g)*Pr*MP.R

which imply:

Pk*K/(1-b)= (1+g/r)^-1*(L+R)/b

i.e. the ratio of K to L should increase by 1+g/r so if I haven’t done anything wrong (which I probably have), a high enough g/r ratio could keep wages up after all.

Ok – so if I´m correct in the reasoning above, I do see what you were after.

[nemi]

sloppy again.

It should read:
r*MP.K/Pk=(r+g)*MP.R/Pr

which was what I did in the calculations.

[nemi]

K: "Yup, this is my mental model too. Robots producing tons of stuff can never impoverish anybody. (This *ought* to be totally obvious). It's the monopolization of finite resources that is the killer."

This is like saying that "obviously" no one can be hurt by free trade - which "obviously" is wrong.

Assume a working market. Assume an additional supply of one billion uneducated workers. Assume that you are an uneducated worker -> not equal to profit. (or assume a supply of educated and uneducated workers without their own capital base, and non capital owners will be hurt).

Given market imperfection (such as unemployment benefits and sticky wages) it does not even have to create a Kaldor Hicks improvement. In the case with the linear production function, it would only create redistribution towards the rich and the really poor (that previously was withouta capital base).

That aside, the (given technology) finite recourse base is probably the bigger obstacle in the long run.

[nemi]

Not my day today. With the linear production function the additional supply obviosly dont have an effect on the wage. After today - I will start to think before I write.

[nemi]

With a leontief, the short run effect is simply redistribution.

[Nick Rowe]

nemi: you are doing fine. And making good progress. You are just thinking out loud.

[nemi]

Nick: Thanks

[Ken Schulz]

NR: "Do you eat 100 times as much rice as someone who lives on $1 per day?" No, I don't, of course. I thought that was the point I was trying to make. In fact, if part, or all of my income were redistributed downward, I would expect total demand for consumption goods to increase a bit. As income increases, consumption increases less rapidly, preference for investments moreso. I thought that this much was not controversial, so I didn't cite empirical evidence.

Could extremes of inequality hinder growth? Here's one reference: http://www.imf.org/external/pubs/ft/fandd/2011/09/Berg.htm

NR: "Until 2008, the last problem with the US was underconsumption! Overconsumption maybe."
I thought we put too much money into dot-coms and McMansions. At the time we thought they were investments, not consumption.

[Nick Rowe]

Ken: so what are people doing with the income? Holding cash? Is the income-elasticity of the demand for money much greater than one? If so, just print more money.

[LSTB]

In other words, Krugman should have written the introduction to Asimov's The Naked Sun rather than The Foundation, in which the main character travels to a planet of 10,000 people living on giant robot-serviced latifundia.

Henry George came up with a model like yours in Progress and Poverty and comes to the conclusion that the trespassing class could only survive at the landlords' whims. Of course, once you concentrate land ownership, whether the workers are robots or low-paid humans is moot. I write more about it in my sig, but this is the gist of it.

[Nick Rowe]

LSTB: Interesting. I just read the bit you quoted from Henry George.

One *maybe* small criticism of Henry George: he *seems* (I may be wrong) to just *assume* (implicitly) that all new technology will be (what he calls) "labour saving". It isn't, and doesn't have to be. Some new technology is "land saving" (or "capital saving").

In my models, by supposing straight out that robots are identical to human workers, and that the only technological advance is in making robots cheaper to produce, I am making the same assumption. But I am fully aware that this is a very special assumption. If I stuck my parameter "a" in front of "N" (natural resources) in my model, instead of where I did stick it, I would get very different results. I was deliberately assuming (or trying to find) a nightmare scenario for labour.

[LSTB]

Nick, you're right: George is missing land-saving capital, which is a point Mason Gaffney makes in his study guide. But George is also trying to find the nightmare scenario for labor, and it's fun to see him try before anyone had thought of robots. I'm not sure land-saving capital is relevant if you're positing that robots require the same sustenance as equivalent human workers.

The real-life fear with robots isn't that they're legitimized slavery but that they are cheaper and better than human workers, which creates a situation I can't imagine not benefiting land owners over non-land owners.

[K]

Nemi,

Yup. Right after hit submit I realized should have said "the representative worker" rather than "anyone."

Nick,

"One *maybe* small criticism of Henry George: he *seems* (I may be wrong) to just *assume* (implicitly) that all new technology will be (what he calls) "labour saving""

The way I read George, he was talking about the asymptotic state of affairs. In the limit we can foresee robots dominating labour in all production, ie using less energy than humans to transform the same inputs (land) into products. But we cannot foresee them functioning without land or energy. In that limit the workers will earn nothing. Why would land owners waste any land/energy on workers, except as he points out, for charity.

In the end land can produce without workers. Workers can't produce without land.

[Frank Restly]

Nick,

Starting with robots, labor, and consumer goods

dR/dt = L * %L + R * %R
dC/dt = L * ( 1 - %L ) + R * ( 1 - %R ) = k * dL/dt

R = Robots
C = Consumer Goods
Y = Total Goods
L = Labor
k = Constant multiple of labor force equating growth in labor force to growth in consumer goods
%L = Percent of labor force tasked to building robots
%R = Percent of robot force tasked to building robots

dR/dt + dC/dt = L + R

R = a * exp ( b * t )
C = ( 1 - a ) * exp ( b * t )

Y = R + C = exp ( b * t )

dR/dt = ab * exp ( b * t )
dC/dt = ( b - ba ) * exp ( b * t )

dL/dt = ( b - ba ) * exp ( b * t ) / k
L = ( 1 - a ) * exp ( b * t ) / k

dR/dt = L * %L + R * %R
ab = ( 1 - a ) * %L / k + a * %R

k * dL/dt = L * ( 1 - %L ) + R * ( 1- %R )
b - ab = ( 1 - a ) * ( 1 - %L ) / k + a * ( 1- %R )

b = ( 1 - a ) / k + a = a * ( k - 1 ) / k + 1 / k

a * [ ( 1 - a ) / k + a = ( ak - a + 1 ) / k ] = ( 1 - a ) * %L / k + a * %R

Rearranging terms and solving for a:

a = [ k * %R - %L - 1 + sqrt ( ( k * %R - %L - 1 )^2 - 4 * %L * ( k - 1 ) ) ] / [ 2 * ( k - 1 ) ]

b = [ k * %R - %L + 1 + sqrt ( ( k * %R - %L - 1 )^2 - 4 * %L * ( k - 1 ) ) ] / [ 2 * k ]

"That sounds nightmarish, right? Because robots will get cheaper and cheaper, and drive down human wages?"

Only if you are trying to maximize robot production ( R ) will robots drive down human wages. If instead you are trying to maximize Y = R + C, then what %L and %R should you use ( hint maximize b )?

What you should find is that the percentage of labor dedicated to building robots ( %L ) should be 0% to maximize Y. The percentage of robots dedicated to building robots ( %R ) should be 100% to maximize Y. Even if robots can build consumer goods faster than the labor force, tasking robots to consumer goods will lead to a lower Y.

The obvious question is - What value does a robot that only builds other robots have?
And the answer is - Ask a robot.

[Magpie]

Just I tiny little question regarding the Land and Robots model (I skipped the second one):

So, if one assumes that robots are perfect substitutes for workers and they earn the same wages (!!!!!)... what is it that makes robots, well, different from humans? I mean, other than Prof. Rowe's word.

Just sayin'!

[Jodie Mitchel]

Howdy! Do you happen to have any blogging education or it is a completely natural talent of yours? Thanks a bunch in advance for your answer.

[Benjamin Adams]

This discussion has gotten me thinking about the political dimensions of robots. Here's my stab at a model.

Suppose we have two types of people, owners and workers.

The overall production function might look like this:

C + T + (P + D)/a + R + S = K + L

C is owners' share of consumption goods.
T is consumption goods transferred to the workers by the owners.
P is the production of new robots used to make consumer goods.
D is the production of new robots (drones) used for counter-insurgency.
R is the workers' level of insurgency.
S is the workers' leisure.
K is production robots.
L is labor.

Owners choose T, P, and D to maximize C while minimizing R, subject to constraints on K + L.
Workers choose R, S, and L to maximize T and S while minimizing D, subject to constraints on R + S + L.

I have no clue what the equilibrium looks like. The important thing to note is that the utility of each type depends on decisions made by the other type.