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Your "musings" are a puzzle to a reader. But what can the reader learn here?

Well, you are coming from a historical perspective. AK/Solow/Smith/Schumpeterian are all writers with an individual perspective. The perspectives are not congruent.

It takes a lot of time to learn the perspective and ideas of past authors. As a teacher, you are concerned about the efficiency of learning and the efficiency of teaching students.

Your concern about new ideas destroying old ideas reflects your concern that some of the old authors may have presented 'dead end' ideas or perhaps misleading ideas. Learning ideas that should be destroyed would be a waste of time and resources. (On the other hand, weak ideas may be helpful as a contrast to a better idea.)

You close reflecting that your father and you took different paths to earn money. Who was more productive? No answer is offered.

Now it is up to me to fathom the meaning of your musings.

Your comment about working under your MX6 indicates that you are following your father in productive ways. The time spent is writing is also productive although you don't know if the seeds contained in the writing are fertile or if the ground that receives the seeds is fertile. Never-the-less, you repair your MX6 and write. (You are productive!)

And now my closing comment.

It seems to me that historical authors have provided tools (mostly in the form of ideas) that we can mold into a recognizable patterns or pictures. The tools (ideas) are not congruent and can be used to paint either a Picasso or a Rembrandt. The beauty of the final picture seems to be in the eye of the beholder.

A stationary assumption is perhaps less interesting. Instead, we invest more in R&D, develop AI, greatly increase educational efficiency, extend life indefinitely, though there is likely a sizable difference between educational efficiency and R&D efficiency, so R&D subsumes both E and L.

I like it! It seems to be saying the ideal (for maximum production) is for an average person to spend exactly half their lifetime in education (K** = eS/2). Just one nitpick: production is *increasing* returns to K up to K**, and only decreasing returns for K>K**.

Nick: "But my father left school at 16 and started work producing food. While I left school at 26 and started work producing...more schooling. And it's not obvious to me whose strategy is more productive."

My father, who spent 50 years producing more schooling, not infrequently reminds me that, as a university professor, my most important contribution to the world is keeping people off the job market.

This old post of yours Of horses and men had some stochastic thoughts about what would happen to people if technology made them redundant, much the way that horses have been made redundant.

I'm trying to figure out what the different theories of growth imply for the "horses and men" problem. If the ideas are equally distributed across people - or, at least, everyone has some valuable ideas, e.g. Smith - then everyone is in some sense useful. But what if patents/intellectual property allow just a few people to get the benefits of all the ideas?

Hope the MX6 is running smoothly - significant progress made here with more mundane household chores.

Ideas (i.e. software in brains) seems like a problematic special case. Humans are poor vehicles for software, thus the model has to account for complex costs and tradeoffs.

Reproducing software on computers is much simpler and cheaper. (There are issues of compatibility and infrastructure but these arise for humans also, are not in your model and probably would have even more negative effects there.)

While the cost is not zero is it very low. Copying the content of Wikipedia or the Linux repository takes a few seconds and the copy is (almost certainly) perfect. The cost is extremely low.

As skills get automated, copying software dominates human learning by immense amounts. For example, imagine five years into the deployment of robot cars, some improvements to the software are made that cut accidents, and/or gas consumption, and/or travel time by 5%. The improvements can be distributed at a cost that is down in the noise. But imagine trying to retrain all the human drivers with similar improvements. The cost boggles the mind, not to mention that the effort would be largely wasted and cause a political uproar.

I think mainly what your model implies is that we should get humans out of the loop as quickly as possible.

Nick,

"(And anyone could criticise the model for ignoring the fact that knowledge may itself be a consumption good.)"

Some (e.g. Bryan Caplan's upcoming "The Case against Education") would argue still further that education is for the most part a positional good, where individual benefit is derived mainly from getting ahead in the signalling arms race on the job market.

In this case, K (understood as human capital that actually matters for productivity) is not a linear function of E, but instead a quickly saturating one. Since both Smith and Schumpeter are right, this by itself isn't bad news. What is bad news is that E is presently way above optimum for essentially the same reasons why spending on ever more elaborate periwigs in the 18th century was way above optimum.

I'll have to think about this (with my own stochastic thoughts) but the quick impression is that you're try resuscitate/reintroduce the classical confusion between capital and technology. Sort of playing fast and loose with A and K and the roles they play in the different models, sometimes pretending they're the same and sometimes pretending they're different. So here's a different set of characterizations:

AK - fundamental confusion between technology and accumulable capital. The source of confusion, the key distinction, is actually not the accumulability of capital, we can have that with technology, but the lack of diminishing returns to it or in other words its rivalness (if you want to get precise, lack of it at "infinity", one of the Inada conditions). Part of the reason why this model confused capital and technology/ideas was because it was more or less developed by quasi, or actual, or folks writing in the shadow of, Marxists, so it inherited Marx's confusion of that aspect.

Solow - No, no, no! Solow just says there's diminishing returns to capital but not technology. And it says, ok, we don't really understand where ideas come from so let's put that aside for now and think about how much we can explain with the rival-diminishing-returns-having capital alone. And it turns out not much. But let's go with your interpretation. Ideas are hard to learn. But how hard they are to learn is just a constant. Then it doesn't matter. It's just a constant hanging around in a Cobb-Douglas production function. Whether you have (A^a)*(K^a)*(L^(1-a)) or A*(K^a)*(L^(1-a)) doesn't matter. We're measuring "ideas" in terms of output-equivalents. There might be some model where the key insight is that ideas can be difficult to learn but it's not Solow.

Schumpeter - My understanding is that Schumpeter is not about how ideas affect the production process (whether they "destroy" old ideas or not) but about the incentives to invent new ideas. The "destruction" in "creative destruction" is not the destruction of old ideas ("standing on the shoulders of giants" and all that) but about *firms* with new ideas destroying firms with older ideas. You can have really new ideas which are a small twist on an old idea but there's still destruction. Microsoft is (was?) sort of famous for that. Lots of destruction in terms of other firms, all the new ideas were small (but non-trivial) improvements on what came before.

" key insight is that ideas can be difficult to learn but it's not Solow" - that should be "more and more difficult to learn". The difficulty of ideas has to increase with the number of existing ideas rather than just be some constant.

As to the mathematical example, consider this instead:

Output is Cobb-Douglas, Y=(K^a)*(L^(1-a)). I hope we can all agree this has diminishing returns to capital, yes? Ok, but in competitive equilibrium w=(1-a)*(K/L)^a, and r=a*(K/L)^(a-1). And we're at steady state, so w/r is constant. Plugging those in we have Y=constant*K, hence there's no diminishing returns to capital! Whoa!

What you have is a production function with increasing returns to S. Let's do it this way. Let the proportion of time spend on education be u. So E=uS. So L=(1-u)S. So Y=eEL=eu(1-u)(S^2). So all you have is a production function which has increasing returns to scale in "total lifetime" (you could make this be just total population). Everything else is just relabeling of the members of the alphabet to make it *look* like there's decreasing returns to scale.

But there's something to this and I'll try to clarify it later. For now, consider an alternative model, the "classic" learning by doing model. For simplicity let's say Y=A(L^a) where a<1, A is technology and L is labor. Labor is fixed. Diminishing returns to labor, right? But A depends on how much output L produces/produced. Let's say the learning happens very fast (instantaneously in the limit). So A=L^b. Plugging in we have Y=L^(a+b) and as long as a+b>1 we now have increasing returns to labor. So which is it? Whatever it is I think it's a bit confusing to label the increasing returns version, L^(a+b) "macro" and the decreasing returns version, A(L^a) "micro" (if that was your intent). They're both macro (I'm assuming a single firm and an inelastic supply of labor, just because I can)

Or too quickly sum up (I apologize for multiple comments, this is the stochastic part of the thoughts), if Y(X,other stuff) is output, diminishing returns to X means that d2Y/dX2<0, holding other stuff constant.

Lord: growth models may or may not have a stationary equilibrium. I think the one I sketched above will, but it's incomplete without some sort of saving function.

Jeff: "diminishing marginal returns" (to be more precise) refers to a negative second derivative.

Frances: the thing I really missed in that old post on horses and men, was land. It's there implicitly, but not explicitly. If land wasn't scarce, horses would roam free in large numbers. It's competition between people and robots to work the land (natural resources) that is the issue.

Was getting vibration under braking (OK, "warped" rotors), but also slight vibration under just engine braking too (puzzle). Also slight knocking over bumps (aha! worn balljoint, which might have been causing the vibration under braking too). So replaced lower control arm (with balljoint, which was worn), tie-rod end (steering, very slightly loose), front brake rotors and pads, then re-aligned steering by trial and error. It's now running fine. It's good for the soul to do real (non-school) work occasionally, even though it probably took me twice as long as a real mechanic with air tools, a lift, and an alignment machine.

Jared: "Reproducing software on computers is much simpler and cheaper."

I think you might have nailed it, in terms of practical implications for the near future.

Vladimir: I suspect that you/Brian are at least partly right, on education being a positional good. But maybe a positional consumption good, as well as a positional production good. (Even if you know it won't help you get a job, you feel good about being as or more educated than others).

notsneaky: if technology just falls out of the sky into our heads at zero cost, it's not capital-theoretic. If it's costly to invent new ideas, then new ideas are capital-theoretic (present costs and future benefits), but old ideas aren't. If it's costly to learn new and old ideas, and lives are finite (so kids have to learn and their learning depreciates when they die), then both new and old ideas are capital-theoretic.

I think that's right.

But yes, I'm playing fast and loose with all 4 sources. Don't let a historian of thought see this!

Off-topic: where is Mark Thoma? I'm worried. He almost never misses a day. Hope he's just taking a very well-deserved break.

Output is food, which comes into varieties, apples and oranges. Apples and oranges are perfect substitutes in the production of food, so F=A+O. Both are produced with labor, with one unit of labor producing one unit of either, A=L(A) and O=L(O). I have 10 units of labor, so A=10-O. Hence F=10. Always. Hence neither apples nor oranges contribute to production, hence neither of them are "factors".


Or in your example. Suppose that S always gets divided between E and L in constant proportions (intertemporal preferences make it so or that's just the nature of education technology). And then somehow e goes up. So K goes up and L doesn't change. dY/dK>0 and d2Y/dK2=0 so no, there's no "decreasing returns to K at the macro level"

There's two logical ways to talk about diminishing returns or returns to scale here. Either you take your derivatives keeping the other "factor of production" constant, or you go ahead and plug everything in and only talk about dY/dS. Anything else is confusing apples and oranges (or actually apples and labor).

Mathematical rest area.
Education may partly be a situational good. So is wealth. Throughout history, elites consistently favored having all of not much to some part of a very big pot, even though their amount would be larger. Their frantic efforts to turn the 20th century western world back to haitian level is par for the historical course.
So, when they tell me that the proles are past the optimum educationnal level, they might as well tell me that the poor have too much wealth. Meaning the poor don't have enough.

I assure you that my idea of calling them "stochastic thoughts" can't be stolen as it is non-exculdable, public domain, un-trademarked TM,free use,batteries not included, no animals were injured in writing this comment and other stochastic disclaimers.

My view of diminishing returns comes from software/tech.

When you have declining marginal costs of production or, in the case of software/digital media, zero marginal costs, you end up making very large fixed investments in order to win the market. You still have diminishing returns to your fixed investment, though. E.g. at some point, you're hiring boutique designers to make the shading on one of the buttons in your "User settings" window look really appealing and you know you've reached that point of low marginal returns per dollar invested. In this sense, investment in IP is no different than investment in rival goods. You can only drop so much money on a neighborhood before the investment opportunities dry up and the money you spend ends up bidding up the price of land or is just wasted. There are only so many skilled managers and legitimate opportunities available and you soon start building big bridges over little creeks. Each economy has a type of productive eco-system with finite capacity to absorb new investment. You can try to grow the eco-system but it takes time and sometimes has a mind of its own. Innovation is a lot like that. Just because you are spending $100 billion on innovation doesn't really mean you are spending $100 billion on innovation.

But, because those capacities tend to be so much higher than what we actually employ, under normal investment rates, over the long run, it can certainly seem that there is an increasing return to scale. That doesn't mean that this property is reliably exploitable. It does mean that there is no competitive equilibrium.

notsneaky: I'm still mulling that one over. First thought: if we had infinite lives (and perfect memories), it would be different. A one-time investment in learning a new technology would create a flow of benefits forever. If there were no costs of learning, but only a cost of research, it would be even more different: a one-time investment in research *by one person* would create a flow of benefits for all people forever.

Costly learning plus finite lives does seem to make a big difference.

Jacques Rene: I'm wondering if it isn't the elite, rather than the proles, who are in danger of having too much education! (But these are heretical thoughts, for those of us in the education business!).

Robert: thanks! It's still a lovely name.

rsj: your comment reminds me of Ricardo's distinction between the intensive margin (adding more labour to existing land) vs the extensive margin (adding labour to the worst land at the margin of cultivation, and pushing the margin of cultivation further out).

It would seem to me that most new ideas nowadays are not even improvements, they are merely rearrangements of existing ideas in ever increasing layers. Division of labour, economies of scale and globalisation increase the need for coordination of the moving parts just to keep the whole rube-goldberg economy running. What's growing isn't so much output as management knowledge over how best to put the pieces of divided labour together. Logistics? This increases overhead while simultaneously reducing slack in production. It also creates a strong incentive not to question underlying processes - that is not to come up with new ideas! - as this could bring down the system with it.

This might be good news for pacifists, because if Tom, Dick and Harry are hopelessly intertwined through their shared output (and educated enough to know about it), they're less likely to do each other in. On the other hand, increasing complexity is also an ideal breeding ground for beaurocracy and inertia. The world becomes more rules-based as interfaces are standardised. But then changing the rules also becomes ever more difficult. There might be a Minsky hidden in there somewhere?

Anyway, if you ask is all this learning worth it?, you must tell us what kind of ideas you're talking about And for us learners to come up with new ideas, we must first understand the world as it is, which is encreasingly complex. Also we must decide what kind of a world we would like to live in, and then figure out the best way to reengineer the world without too much collateral damage.

So to tie in with your examples above, AK and Smith are adding new dots without considering that they also have to be connected. Each new dot creates the need for a lot of new lines. Solow doesn't recognise that it might be worth doing things better without ending up with more things altogether and Schumpeter seems firstly a bit hap-hazard for us risk-adverse administrators of inertia. Secondly, it would seem that many competing ideas can coexist, even within the same system. So although not everyone must learn all old ideas, it still takes someone to coordinate between people who have different ideas about how to achieve the same goal.

There's two different issues being conflated here. One is whether or not there are diminishing (usually use "decreasing" for returns to scale and "diminishing" to returns to a factor) returns to ideas in the real world. I have no ... idea (well, a little bit of one). The other one is whether in the model you set up there are diminishing returns to ideas. I'm saying, no, in your model there are no diminishing returns to ideas. There are constant returns to ideas + plus a resource constraint. That's it. Or, under another interpretation, I'm saying it doesn't make sense to talk of diminishing - or any other kind - of return - to ideas because all we can really talk about is returns to S, time available.

Now your model may perfectly well be a very fine model but whatever it's results and implications are, these are not due to "decreasing returns to ideas".

One other, related, comment. We need to distinguish between the production of the final consumption good, and the production of a factor, in this case the set of ideas. You can have no diminishing returns to ideas in production of the consumption good - because them ideas are non-rival, but at the same time you can have diminishing returns to ideas in the production of ideas; when it becomes harder and harder to invent new ideas, once a lot of them have been invented, a sort of a "idea exhaustion" story (possibly being offset by a different effect, the "standing on the shoulders of giants"). And that's sort of similar to the story you're trying to tell, I think. And it's also actually how you get some of the "New Growth Theory" models to have a steady state (otherwise the growth rate of new ideas would explode)

Oliver: even businessmen thought that in 1914. The economic disruption in August 1914 was such that a lot of the "lads battallions" were made up of newly unemployed workers,laid-off because of the trade disruptions. But there was still enough slack capacity (mainly in the form of at-home women) to supply munitions for the carnage.
So maybe a benefit of having women working is to remove the slack necessary to finance a world war.Free daycare!

What happens when machines start creating new ideas?

"it probably took me twice as long as a real mechanic with air tools, a lift, and an alignment machine"

OT: No air tools! That's brave. I changed the shocks on my '06 Mazda6 with nothing but hand tools. I broke one bolt with the breaker bar and had to cut two others with a cut-off wheel. Hopefully the anti-seize I applied helps for next time, but I think I'll try one of those new electric impact wrenches. I also wimped out and took it to the shop for an alignment.

Jacques:

Free Daycare indeed! But I was talking about slack in production processes, i.e. productivity, not unemployment or underemployment. I was merely trying to point out, in an admittedly long and winding way, that seemingly diminishing returns on ideas are a not a consequence of ineffective ideas at the production level but of our incapability to organise ourselves in ever larger and interdependent units. I do wonder whether we will manage to simplify things without having to de-coordinate in a messy way.

Patrick: I bought an electric impact wrench at Canadian Tire, and it's useful, but it won't work on the big really stuck bolts. Not enough torque. But I have (eventually) managed to remove almost every bolt with a longish breaker bar plus cheater pipe slipped over the end (or a wooden lever levering the end of the breaker bar against the ground), plus maybe a blowtorch. I failed on an axle nut, where the breaker bar broke when I was jumping on the end. But it does wear my body. I bought a cutoff wheel, but am mostly scared to use it. Spent 30 minutes using a hacksaw to cut off the swaybar endlink, because the bolt was spinning with the nut, and the allen key wouldn't hold the end of the bolt. Canadian winters are the problem.

notsneaky: the assumption I'm (implicitly) making in this model is that ideas and learning are (strict) joint inputs, so the marginal product of ideas alone is undefined, and we have to define a composite input of ideas+learning (ideas that are actually in workers' heads rather than in books somewhere), and we can only talk about the marginal product of that composite input. Which I think is what's causing the problem here, because when we talk about the MP of that composite input, what precisely is the other input we are holding constant? Total labour? Or labour that is directly producing the good, rather than learning the idea?

I'm writing a post arguing that ideas and learning must be joint inputs. But now I'm not sure I'm right. Maybe I will just change the title to a question, and post it.

Nick, let's think of it in yet another way. Forget about production, factors of production, or ideas. Think about utility.

My utility function is U=xy. Does this utility function have diminishing utility of x? Hell no. Ok, but when I maximize this utility function I'm faced with the budget constraint x*px+y*py=I, where I is my total income. Solve that budget constraint for y; y=(I/py)-x*(px/py). Plug the budget constraint into my utility function; U=x*((I/py)-x*(px/py)). Now take the derivative of my utility function with respect to x, AFTER (for some reason) you've plugged the budget constraint in; dU/dx=whatever. And the second derivative d2U/dx2=whatever<0. Whoa! Now my utility function has diminishing utility of x! How can it be both? Contradiction. Or more precisely, there's a conceptual mistake in there. Which is the mistake you're making.

All of this has absolutely nothing to do with the question of whether ideas, or learning, or capital, or "x", or "K", or "E" actually have diminishing returns in the real world, how ideas are adopted, whether they have to be learned, whether there will be no more ideas invented once Bill Gates passes to his grave. Etc. What it has to do with is doing things correctly. You can't have a concept which has two contradictory properties. If you do, it means you've not defined your concept properly ("diminishing returns") and you're equivocating somewhere, using the same word - or even the same mathematical symbol (this IS actually a prime example of "Mathiness"!) to mean two different things.

Also, what are "joint inputs"? That is not a snarky question, I honestly don't know.

notsneaky: "Also, what are "joint inputs"? "

See latest post. Inputs that are strict complements, so Isoquants are L-shaped. A worker without a shovel is useless; a shovel without a worker is useless. They are used in fixed proportions.

Ah, perfect complements. So what you have above there is really K=Min[eE,X] where X is the stock of existing ideas, the maximum number of ideas that can be potentially learn'd. I still say that it's incorrect to say that in this case there are diminishing returns to ideas. But I'll read the next post.

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