[Update: anyone who doesn't get my point, or who wants to see the point explained by someone who understands math modelling better than me, or who wants to see it from the perspective of a real scientist (not an economist) who studies predator-prey dynamics, should read Jeremy Fox .]

Instead, I am going to use the words "linear" and "non-linear" in the way Artsies use those words. A "linear" plot is one that starts at the beginning, then continues with day causing following day, until it gets to the end. A "non-linear" plot curves back on itself, with simultaneity, flashbacks and flashforwards, and self-referential asides where the characters wonder if they are in a story and how it's going to end.

When economists say that something is "linear", rather than "non-linear", they normally mean it is a straight line, rather than curved. Y=a+bX is linear; Y=a+bX 2 is non-linear.

The very simple story of demand and supply, as we teach it in ECON1000, is a fundamentally non-linear story. If you try to tell the demand and supply story as a linear story, you always get into deep trouble. "Bad weather causes a decrease in supply, which causes an increase in price, which causes a decrease in demand, which causes a decrease in price, which causes a decrease in supply, which causes an increase in price...."

Explaining demand and supply to first year students is difficult, precisely because it is a non-linear story. They want to know whether a change in quantity causes a change in price, or whether a change in price causes a change in quantity? And in which direction will price change if quantity changes, or will quantity change if price changes? And you can't answer that question, because it doesn't make sense. We can only answer "both".

And so we draw a supply and demand curve diagram, point to the intersection of the two curves, and say that price and quantity are co-determined simultaneously in equilibrium by the supply and demand curves. Where the supply curve is the whole relationship between price and quantity supplied; and the demand curve is the whole relationship between price and quantity demanded. And explain that supply, quantity supplied, and quantity sold are three different concepts. And explain how demand, quantity demanded, and quantity bought, are also three different concepts. And tell a story of the disequilibrium sub-plot in which an excess supply causes price to fall to the equilibrium, and excess demand causes price to rise to the equilibrium. And distinguish that disequilibrium sub-plot from the main equilibrium plot in which shifts in supply or demand curves cause the equilibrium price to rise or fall.

"Equilibrium" is economists' code-word for "this is a non-linear story".

The very simple standard story of demand and supply is a non-linear story. That's what makes it so hard to teach properly. Students really really want to hear a linear story. They like linear stories, because linear stories are so much easier to understand, remember, and explain. You can even write linear stories in point-form. "S up arrow P down arrow D up arrow Q up" would say it all, if the plot were linear. The "arrows" indicate "precedence", in both the causal and temporal sense. If the plot were linear, we could easily lay out the concrete steps through which an increase in supply caused a decrease in price and an increase in quantity, taking one causal step at a time. But it's not linear.

My car has a linear transmission. The engine turns the main shaft, which turns a cog, which turns another cog, which turns the auxilliary shaft,..which turns the axles, which turn the wheels, which make the car go forward. The monetary policy transmission mechanism, even in the very simplest and crudest version of that story, is non-linear. Even calling it the "monetary policy transmission mechanism" is a highly misleading metaphor, because it leads people to expect a linear story.

The simple story of monetary policy I teach in ECON1000 is non-linear like the demand and supply story, only more so. That's because there are more goods, more supply and demand curves, more endogenous variables that get determined simultaneously, and almost everything depends on almost everything else.

I have a shameful admission to make. A couple of times, after some students still didn't get it, I broke down. I told them that this wasn't exactly right, and then wrote down: "M up arrow r down arrow I up arrow Y up arrow P up". And the students loved it. Yes! This they could really understand! They all wrote it down in their notes, memorised it, and regurgitated it on the exam.

What I did was wrong. I linearised a non-linear story. But that's not the point of this post.

The point is that the students really really wanted a linear story of the monetary transmission mechanism. And when I gave them a linear story they thought they understood it. They didn't. They misunderstood it. They misunderstood it just as badly as a student who thinks "an increase in supply causes a decrease in price causes an increase in demand causes an increase in quantity" misunderstands demand and supply.

My students were like the people from the concrete steppes. The people from the concrete steppes want a linear story of the monetary policy transmission mechanism. Sorry, but you can't have it. If you think you understand monetary policy through a linear story, you don't understand it. The true story is non-linear. There is simultaneous causation, so endogenous variables are co-determined in equilibrium. Expectations matter, so the story has flashforwards and flashbacks, where the future affects the present. Those expectations are self-referential, because the characters in the story know the author who works at the central bank, and can anticipate the future plot he plans to write, which affects their actions today. That means a loosening of monetary policy could mean that interest rates rise, not fall. You cannot reason from an interest rate change (Scott Sumner), because interest rates are endogenous variables.

The people from the concrete steppes find this all very frustrating. They want a story with a linear plot, because a linear plot is what they understand. And if they think the linear plot cannot work at the Zero Lower Bound on nominal interest rates, they conclude there must be no plot in which monetary policy works. And they think that any non-linear story is just handwaving and mystification.

Here is a very simple non-linear story, which the people from the concrete steppes won't like. We know that for any equilibrium time-path there exists a second equilibrium time-path along which all nominal variables (like NGDP, and the money supply) are higher or lower. This means that a permanent loosening of monetary policy will eventually move the economy towards one of those time-paths along which all nominal variables are higher. This means that the expectation of a permanent loosening of monetary policy will mean the expectation of an eventual permanent increase in all nominal variables like NGDP. (This is how monetary policy always works, regardless of the ZLB, because interest rate changes are an endogenous symptom of monetary policy changes, not an integral step in a linear causal chain.) An increase in expected future NGDP will increase demand today, and increase current NGDP. It would really help people to expect a permanent loosening of monetary policy and a permanent increase in future NGDP if the central bank announced a commitment to a higher future NGDG target, because hitting that target would require the central bank to loosen monetary policy permanently.

Not very linear, is it? But then economics isn't a linear subject, and hasn't been for hundreds of years. So what did you expect?