Failed Singularity Model
Noted Yale economist William Nordhaus has a new paper “Are We Approaching an Economic Singularity? Information Technology and the Future of Economic Growth”:
Assume that labor is constant, that all technological change is capital-augmenting at 10% per year, and that the elasticity of substitution between labor and information capital is 1.25. Figure 3 shows a typical simulation of the share of capital and the growth rates of output and wages.
In this model, capital slowly gets a larger share of total income and the economic growth accelerates, even though the rate of innovation never changes. Nordhaus lists six empirical predictions for the sign of observed parameters, and finds that four of the six are rejected by our best estimates having the opposite sign. And this doesn’t include the fact that our best estimates find the elasticity of substitution between labor and capital to be less than one. The two sign predictions that match the data suggest it would take a century or more before growth rates exceed 20% per year. Nordhaus says, “The conclusion is therefore that the growth Singularity is not near.”
Of course this is far from the only possible economic model of a singularity. But it sets a good standard for future efforts. Can anyone find a concrete simple economic model of singularity that better fits the data?
As said below, I don't think the doubling can't continue forever, and just because it's been steady for the past decades doesn't mean it's going to be steady for 20 or 30 more years. If the innovations you mention below fall through on delivering the effectiveness, we will see a sudden decrease in computational growth.
I also don't think it's clear that at some threshold, the value of a calcualtion per second can't fall by a factor of two every year. Here, a lot depends on how much computation can be used to make up for physical or other hard limits in the economy. If better and better ways to use material and energy are unlockable by faster computation, then the value of computation may not decrease. But if it only forces people, corporations and nations into information-theoretic zero-sum games without solving harder limits on the economy, it may indeed fall more than a factor of two per year beyond that point, even if it is counterintuitive.
As for the Fukushima robots you mention below, they could also be handeled by humans with remote control, but I'm sure there are many applications where faster algorithms are a marginal improvement. I just don't think they will necessarily to a singularity kind of event without diminishing returns before a point where we don't recognize the human condition anymore, etc.
As for the historical observation that the economic value of human brains has increased, yes, but that may not be projectible onto the future. For example, in this time we have developed ways to use uranium and fossil fuels that our ancestors didn't have. But those are limited physical resources and there is no certainty that we can repeat this many times into the future. (We also used up other natural capital).
And ultimately of course, there's the question of desirability. Let's say ems are invented. Then we suddenly are back to a low per capita GDP because Malthusianism applies again. Same for any equvalent "innnovation" in reproduction rates. And due to the zero-sum nature of politics and various ideologies, it may be argued no amount of wealth will ever make people truly happy and content. We could all be billionaires, in a material sense, and the world could still be very shitty, without good ways of resolving these conflicts and the general misery of the human condition. :-)
"Do we have any good reason to think it will continue 20 to 30 more years? Most of the gains from miniaturization have already been made."
Well, it's been pretty steady for the past ~70 years. Also, we haven't really developed the third dimension in chips. And neuromorphic (massively parallel, like the human brain) chips are just being developed.
Neuromorphic chips
"Of course, even if a "brain equiv" chip costs more than a brain, it might be programmed better to do the range of things humans are bored to do."Or things that humans are justifiably afraid to do. Imagine how much money could have been saved if robots had swarmed onto the Fukushima Diaiichi plant, and provided seawater such that none of the explosions would have occurred.