Nothing new here. Think of this as a teaching post. Partly for the people of the concrete steppes, but for others too. I'm just going to talk about different ways that expectations can affect what happens. The main distinction is between multiple equilibria and unique equilibrium.
1. Multiple Equilibria
Here is one simple model of multiple equilibria: Y=E(Y)
Let Y be something that people do. Let E(Y) be people's expectations of what people do. The simple model Y=E(Y) says that people do whatever they expect other people to do. Expectations are totally self-fulfilling. People can expect anything they like, and it happens, just as they expect. There is a very large number (a continuum) of equilibria. Anything whatsoever can be an equilibrium. Or almost anything, if Y=E(Y) is only true over some range.Here is another simple model of multiple equilibria, but where not everything is an equilibrium. Y=[E(Y)]2 + X, where X is some exogenous variable. This model has two equilibria. For any given level of X, there are two different levels of Y at which expectations will be self-fulfilling. But if people expect some other level of Y, their expectations will be falsified by their actions. And if X changes, both equilibria will change too.
It is easy to construct a model with three, or four, or any number of equilibria. Depending on how people form their expectations, and how they adjust their expectations if they turn out to be false, some of those equilibria may be unstable, and others stable.
Examples of multiple equilibria are all around us.
I use the word "cat" to mean cat, because I expect other people around me to to do the same. (But when I'm talking to some other people around me, I use the word "chat" to mean cat, because I expect those people to do the same). There is no reason why we couldn't have a different equilibrium in which we all used "cat" to mean dog, and "dog" to mean cat. Almost any word could mean cat.
Another example is which side of the road to drive on. Here, Y can only take on two values: "left" or "right". If you expect everybody else to drive left, you drive left, and if you expect everybody else to drive right, you drive right.
Most people obey the law and follow the leader because they expect everybody else to obey that same law and follow that same leader. If everybody thought the law was a different law, or the leader a different leader, they would obey that different law and follow that different leader. The law is whatever people expect to be the law. The leader is whoever people expect to be the leader. Many different things (though perhaps not anything) could be the law, if people thought it was the law. Many different people (though perhaps not anyone) could be the leader, if people thought that person was the leader. The constitution, which is supposed to tell us who gets to make the law and who gets to be leader, wouldn't be the constitution if people didn't think it was.
2. Unique Equilibrium.
Here is a simple model with expectations and a unique equilibrium. Y=E(X). It is not X that determines what people choose to do; it is people's expectations of X that determines what people choose to do. It is not rain, but our expectations of rain, that determine whether we carry an umbrella.
Here is a slightly more complicated model. Y = 0.5E(Y) + 0.5X. What people choose to do depends partly on some exogenous variable X, and partly on what they expect other people to do. But there is only one level of Y, given X, at which people's actions will confirm their expectations of others' actions. Most static economic models are like that.
Now lets complicate that model slightly, by introducing time. Y(t) = 0.5E(Y(t+1)) + 0.5X(t). What people do today depends partly on X today, and partly on what people expect people to do tomorrow. Most dynamic economic models are like that. And you need to say something about how people form their expectations in order to solve those models.
One assumption is rational expectations. People's expectations are consistent with the model. Taking that same equation and leading it forward one period gives you Y(t+1) = 0.5E(Y(t+2)) + 0.5X(t+1). If people expect that that is how people will choose tomorrow, then E(Y(t+1)) = 0.5E(E(Y(t+2))) + 0.5E(X(t+1)). We can substitute this solution for E(Y(t+1)) into the first equation, repeat again and again, and get the equilibrium as:
Y(t) = 0.5X(t) + 0.52E(X(t+1)) + 0.53E(X(t+2)) + etc.
[Somebody please tell me politely if I've screwed up the math, as I usually do.]
The solutions to almost all modern macroeconomic models look like that. If X is some policy variable, then it is not just current policy that matters, but people's expectations of the whole path of future policy.
If you make a different assumption about expectations you will get a different solution. But if people think that policy affects what people do, and also think that expectations affect what people do, you will still get some sort of solution which looks something like that.
In that example, I used "0.5" as a numerical parameter value for both E(Y(t+1)) and X(t). That means that Y(t) is determined 50-50 by current policy and by expected future policy. If I had originally measured time in years, and then switched to measuring time in 6-month periods, I would need to change those parameter values, and make the model (roughly) Y(t) = 0.75E(Y(t+1)) + 0.25X(t). And the solution would now become:
Y(t) = 0.25X(t) + 0.752E(X(t+1)) + 0.753E(X(t+2)) + etc.
[Damn. i've screwed up the math there, haven't I. I need more coffee, or some help.]
Now Y(t) is only 25% determined by current policy, and 75% determined by expectations of future policy.
As we shorten the time-period still further, we approach a solution in which Y(t) is 99.99% determined by expectations of future policy.
3. Counterfactual conditional expectations matter too.
I do not break the speed limit. (OK, just assume I don't.) I do not expect the police to give me a speeding ticket. But I do expect the police would give me a ticket if I did speed. Which is why I don't speed. So in equilibrium I do not speed. And I don't expect the police to give me a tcket.
What people do in equilibrium is in part determined by people's counterfactual conditional expectations of what other people would do away from the equilibrium.
4. Monetary policy at the ZLB.
New Keynesian macroeconomic models have solutions that look like:
Y(t) = 0.25X(t) + 0.752E(X(t+1)) + 0.753E(X(t+2)) + etc.
4.1 For quarterly data, it's a smaller number than 0.25 (and so a bigger number than 0.75) and so expected future policy matters a lot more than current policy;
4.2 The policy variable X(t) is interpreted as a nominal interest rate, but the nominal interest rate is not exogenous. It is determined by a feedback rule so that it depends on Y(t). This means that people's expectations of policy can be described [I should have said "and must be described"] as beliefs about the parameters in that feedback rule, rather than expectations about the nominal rate of interest itself. In all New Keynesian models, if the central bank announced and stuck to a time-path for X(t) that was not a feedback rule, (and so did not depend on Y(t)), the model would have multiple equilibria, with nearly all of those time-paths for Y(t) leading to an explosive or implosive path for the inflation rate. (If expected inflation were to increase, that would reduce the real interest rate for any given nominal interest rate, which would increase demand for goods, which would increase inflation, which would increase expected inflation, and so on.) What prevents those explosive or implosive paths happening are people's expectations about what the central bank would do if one of those paths were to happen (it would increase nominal interest rates by enough to stop it happening). This means that counterfactual conditional expectations matter too in New Keynesian macroeconomic models.
Even if you think about monetary policy as New Keynesians do, as setting a counterfactual conditional feedback rule for the nominal interest rate, this does not mean that monetary policy cannot be loosened at the ZLB. If people expect that at some future time, however distant, the economy will escape the ZLB and the central bank will want to raise interest rates to prevent the inflation rate rising too much, monetary policy can still be loosened today. The central bank can commit to delaying the time at which it would otherwise increase the nominal interest rate. It could commit to keeping interest rates "too low for too long". A better way to communicate that commitment might be to commit to a higher price level or NGDP level in the future feedback rule for monetary policy. And it might be better still to stop talking about monetary policy in terms of interest rates, and describe the monetary feedback rule in terms of a setting for some asset price as a function of an NGDP target.
I don't have great hopes for this post persuading people to change their perspective on expectations. But one can only try.