I've always found that learning new areas always goes a lot better if you start with a key insight of what the field is about. Often this is not presented or explained at the beginning of the course, and you have to deduce it later on.
For instance, I would have better grasped the epsilon-delta definition of a limit if the instructor had started with something like:
- Our intuitive definition of a limit is that as we get closer to this point, the function gets closer to this value. It has turned out to be very tricky to formalise this intuition, however. Early mathematicians used calculus without a good definition of limit, and their informal definitions led to a lot of paradoxes. The epsilon-delta definition is a bit clunky and may seem counter-intuitive, but it actually manages to capture our intuitive definition without paradoxes and problems - that's why we choose it, not for its elegance (though you will come to appreciate it). With that in mind, let's have a look at it...
Similarly, I would have made more rapid progress with Gödel's theorems if, before giving the formal definition of Gödel numbering and of the provability symbol □, someone had clarified that direct and indirect self-reference was a problem. If a formal system of a certain complexity can talk about its own structure, even without "realising" that it's doing so, problems will arise. Some of my other key insights in the field can be found in my post here.
So when I do stumble upon a key insight, I want to share it. I've found some recently in Keynesian economics, giving me a much better grasp of what makes that economic theory tick, and which would be my point of entry should I ever study the subject in detail. The two key insights are:
- Keynesian models do not require irrationality. Unemployment can persist (in the model) even if every agent is completely rational.
- Hence theoretical macroeconomics really is different from theoretical microeconomics.
Of course, Keynesianism makes great use of irrationality or partial rationality of the agents (such as the stickiness of wages or the irrationality of bubbles), but it was a revelation that rational models, full of Homo Economicus, could still produce excess unemployment.
This seemed intuitively very odd. After all, if there is unemployment, wages should fall, making it more attractive to hire workers. Therefore the equilibrium should be that everyone who wanted to work at the wages available should work. And this is not only an equilibrium, but an attractor: free-floating wages should move the economy towards the equilibrium.
But this lecture presented the rest of the argument. In a closed economy, investment (by firms) plus consumption (by individuals) must be equal to the total production of the economy - you can't sell stuff to thin air. Similarly, the amounts sold by firms translate into income for firms, shareholders and workers - you can't generate income without selling to someone. Over the short term, things can move out of equilibrium (people can increase or cash in their savings), but over the long term it has to balance.
That equilibrium is also an attractor. So we have two equilibrium processes - the wage changes, and the consumptions plus investment equality. Notice, though, that they interact! As wages rise and fall, people's incomes rise and fall, and hence their consumption, which feeds through to the incomes of firms and hence to their own levels of investment and salaries...
The question then, is whether there exists a joint equilibrium for both processes at once (more properly, since consumption consists of many markets, a general equilibrium for the whole economy). We'd want an equilibrium that was also an attractor, since we'd want to move to that state. In some circumstances, such attracting joint equilibriums exist - but in others, they don't.
So, at least in the model, excess unemployment can persist in the presence of fully rational agents.