More random musings, mostly to try to get my own head straight on some questions. (I should steal Robert Waldmann's idea of calling them "stochastic thoughts".) And probably not very original. But sometimes it's easier to try to reinvent the wheel than walk to the library and read all the books on "wheels"; and experiential learning is a good thing anyway, they tell me.
"AK": Ideas (about new technology) are neat, because a good idea can double output across all scales of the other inputs.
"Solow": But the trouble with ideas is that you have to learn them (investment), and that's time you can't spend working at producing consumption goods. And then you die, so that investment is lost (depreciation). So there's decreasing returns to ideas at the macro level, because more ideas to learn means less time producing goods using those ideas.
"Smith": But we have a division of labour, so not everyone has to learn everything. And the division of labour is only limited by the extent of the market.
"Schumpeter": The really good ideas are new ideas that destroy old ideas, so you don't have to learn the old ideas and can torch the ancient libraries.
This is probably empirically important. If you add up all the time we spend at school college and university, plus on-the-job training, plus all the teachers' time, plus all the time parents and their kids spend on the teaching and learning that isn't counted in GDP,.... it's a lot. Maybe even around as much time as we spend working at other things. And time spent Teaching and Learning existing ideas is probably an order of magnitude bigger than time spent on Research and Development of new ideas.
Golden Rule: let's solve for the level of technology that maximises per capita consumption in a stationary economy (where nothing changes over time).
Let the production function be Y=K.L, where Y is output of the consumption good, K is the stock of ideas in workers' heads (technology), and L is labour used to produce consumption goods. That's an Increasing Returns to Scale production function; if you double both K and L you quadruple Y.
Let the production function for Teaching-and-Learning be K=e.E , where E ("education") is time spent teaching-and-learning the existing stock of ideas, and e is a parameter representing the efficiency of education.
People have finite lives. The per capita resource constraint is L+E=S where S is total lifetime. (I am ignoring leisure for simplicity, and there is no R&D in the steady state, by definition.)
That means Y=K(S-K/e), so dY/dK = S - 2K/e , and d2Y/dK2 = -2/e < 0 (did I get that right?), so we have decreasing returns to K at the macro level, once education is made endogenous.
To solve for the Golden Rule level of K, set dY/dK =0, so K** = eS/2 . The level of technology that maximises steady state consumption is an increasing function of human longevity and the efficiency of the education sector. Which makes sense. Technology that increases the efficiency of the education sector would be very important (though the model treats e as exogenous, and it seems not to increase very quickly in reality.)
The dynamics are left as an exercise for the reader, because they are too hard for me, after spending nearly all yesterday under my MX6 with 2(!) socket sets (regular plus impact) breathing liquid wrench fumes (that's my excuse anyway). What would this model's version of the standard Solow diagram look like? We can reinterpret Solow's "d" as the death rate.
Adam Smith would criticise the above model for ignoring the division of labour in learning. If we double the size of the market (by doubling population, or free trade, or reducing transport costs) then we could double K** and hold E and L constant, because each individual could halve the proportion of K** that he learns (each learning a different half of the doubled total).
Schumpeter would criticise the above model for assuming that understanding existing ideas is a prerequisite for learning new ideas. Sometimes it isn't, and the new ideas destroy the old ideas, so we learn the new ideas instead of the old ideas.
(And anyone could criticise the model for ignoring the fact that knowledge may itself be a consumption good.)
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.