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Nick, I gave this post a quick one over, and I'd like to study it more. My first thought was about your last sentence:

"The trouble with Keynesians is........they assume rational expectations and full information. Without even realising it."

Maybe the answer's in the post and I didn't see it, but isn't that true of monetarists too?

O/T: You are probably sick to death of this, but I made my into a blog post, complete with a diagram of the "rectangular hyperbolas" w/ crossing points, etc, illustrating the difference between your solution and Scott's. I hope it's correct.

Well now. There is a difference between "I have written a model in which Keynesian conclusions require rational expectations and full information" and "Keynesians assume rational expectations and full information."

Tom: thanks. looks roughly right on a first glance.

JW: Good comeback! But keynesians want demand to depend on realised sales, and also have realised sales equal demand. There is a bit of a timing problem there, especially in a monetary economy, where the demand for money would always be zero if spending and sales were perfectly synchronised for all agents. The simplest way to get a keynesian model is to use my early/late shoppers device. But when we do that, you can't assume Ye=Y without assuming full information and rational expectations.

Nick: I remember reading in Keynes' GT that demand depends on expected income, I don't have the page number at hand but it was part of the response to an increase in autonomous spending and the multiplier. It may be contradicted elsewhere in the book of course, but it struck me as how different that was to the usual presentation of his demand theory such as your point (to JW) about realised sales.

[This is an aside, I haven't read this post properly - I haven't finished thinking about apples yet.]

HJC: "expected income" can mean two different things, in the context of this model. I use it to mean "expected income in the current period", but it can also mean "expected income in future periods". My guess is that Keynes meant the latter. He assumes agents know their current period income, while at the same time wanting each agent to choose his current expenditure as a function of his current income. Unless all agents know what every other agent is choosing (there's a tatonnement on current expenditure) his assumption doesn't really work.

Even in a continuous time model, where the length of the current period is arbitrarily short, agents buy and sell in discontinuous lumps, that are not perfectly synchronised, which is why they hold money. And if you get an unexpected lump of income, it might just mean it came a few seconds earlier than you expected, so you simply hold the extra money in your buffer stock for a few extra seconds, and do not change your spending plans. There's a temporary/permanent signal-processing problem.

If the following lumps of income continue as previously expected, your expectations of income do not change, but you do hold more money than you had expected, so you get rid of it. And then the hot potato starts getting passed around. My early/late shoppers was just a crude device to illustrate this sort of thing.

I'm trying to see all this in a standard IS-LM diagram.

Suppose everything involved in determining the position of the IS curve stays the same from one period to the next. To begin with, let's assume we have Y = Ye and we're at the point on IS corresponding to the central bank interest rate r*...call this point (Y*,r*). To simplify, let's also assume there's no borrowing from the central bank in this initial situation.

Now suppose there's an unperceived fall in V. Because it's unperceived people have no reason to not expect Y = Y* or adjust their borrowing. Which means the quantity of money will not change and hence it will be insufficient to keep the economy at (Y*,r*) given the fall in V: the ex post LM will be to the left of Y* and the economy will be at some point (Y',r') on IS above (Y*,r*). With Y = Y' < Y*, everyone now will wish they'd borrowed money from the central bank to finance spending beyond Y'...but that's the price they pay for having made this period's borrowing choices based on an erroneous forecast of Y.

If we now make V a random variable whose value only becomes known after each period's borrowing choices have been made we get a model where Y is itself a random variable centred on Y*. Same thing for r with respect to r*, if we think of r as the ex post market interest rate.

Am I on the right track?

Maurice: "I'm trying to see all this in a standard IS-LM diagram."

So am I. I keep drawing IS and LM curves on scraps of paper, and tearing them up.

Start where all 3 curves cross at the same place. Then suppose the bank cuts r, but everyone thinks it must be because the demand function 1 has shifted down, so the bank needs to cut r, so Ye stays the same, but the demand function has not in fact shifted, so each individual increases Yd, but expects every other individual to keep Yd constant.

There's a movement down along the AD (1) curve, holding Ye constant.

Nobody expects the vertical LM to shift right, but it does shift right. It shifts further to the right than Yd increases. The late shoppers find their actual Y exceeds Ye, so increase their Yd still further. The actual vertical LM determines Y.

I think that's right.

It strikes me that Nick's ex-post vertical LM is a measure of lack of information / uncertainty / forecasting error (expectations ~~ forecasting). The ex-ante horizontal LM is not 'wrong', rather it is subject to forecasting error. If forecasting error is mean 0, then we get Maurice's observation "Y is itself a random variable centred on Y*. Same thing for r with respect to r*". Thus the ex-ante horizontal LM demonstrates Box's observation "essentially, all models are wrong, but some are useful". Now perhaps Nick's model is *less wrong and more useful* since it accounts for more of the world.

Reminds me of two days ago when I was explaining to students that our model of 'system availability' must be more sophisticated when we move from co-located data replication nodes to globally distributed nodes. Reality is complicated and defies models with assumed perfect agents (or perfect networks).

Squeeky: I'm still wondering....You could say it's the horizontal LM which is wrong ex post, or the IS. No agent knows how much other agents are going to spend and how much income they will earn. The early shoppers don't find out until it's too late to do anything about it (until next period). The late shoppers do find out, but may wish they had borrowed more money, or less money.

And you do need the horizontal LM and the IS to help figure out where the vertical LM will be. Maybe the two ways of thinking about the world are equivalent. Dunno. Still thinking about this one.

A variant on the model would be where all agents are early shoppers. All must place orders for goods in advance, before they find out how much other agents have ordered. They all put their cash down at the same time, and then there's a short delay before they find out how much cash other agents have put down, but they aren't able to change their order after finding out. The vertical LM curve is then the very simple CIA: Yd=M/P. We lose the within-period Keynesian multiplier if we set it up that way. But the following period, if agents learn that they have earned more income and hold more money than they expected, they will revise their expectations and want to spend more and want to hold more money. The keynesian multiplier and monetarist hot potato process are equivalent, and work themselves out over time, as agents continue to revise their expectations.

"The late shoppers do find out, but may wish they had borrowed more money, or less money."

Is it not that the late shoppers would wish immediately after the coin-toss that they should not have borrowed any money? While the bank is open everyone should borrow the same amount as the group is still homogenous at that point. After the coin-toss, the late shoppers realize they don't need what they borrowed and would return the money (if they could). Once their income falls below what they anticipated they have to make the decision if they only want to spend what they received in income or if they want to supplement their income with the money they had taken out to keep spending as much as they had initially decided on. In the first case, the early shoppers will exactly hold as much as they had taken out in the end, in the latter they will have a surplus and the late shoppers an equivalent deficit.

Something I can’t figure out…

Remove all sources of uncertainty from the analysis and suppose nobody ever makes any forecast errors. Suppose Yd = F(Ye,r) applies to everyone individually and in aggregate (just change the definitions of Yd and Y as required). The central bank reduces its lending rate from an initial r* to r**. We move down the IS from the initial equilibrium level of income Y* to the higher Y** at the point where the Keynesian LM intersects the IS. All this occurs because of an increase in borrowing and the resulting increase in the quantity of money.

Except...look at it from the typical individual's point of view. We have Yd = F(Y**,r**) = Y** at the new equilibrium. If the individual correctly anticipates this why would they ever increase their own borrowing? They can just sit back and wait for the increase to their income that will occur as everyone else increases their borrowing, knowing their income will then be just enough to cover the spending they’ll want to do at (Y**,r**). But if everyone thinks this way we never get to Y**.

On the other hand, if the individual expects no one else to increase their borrowing and income to stay at Y* they'll have a reason to increase their own borrowing when the lending rate falls, to achieve their desired level of spending at the new central bank lending rate. If everyone thinks this way we’ll actually end up at Y**. But then everyone's expectations with respect to their income will have proven incorrect.

In other words, to get to (Y**,r**) and sustain this as the new equilibrium wouldn't you'd need people to incorrectly expect Y = Y* in the first instance and then to make the same forecast error period after period thereafter?

Nick, O/T: Scott brings up your previous post today, and I remind him of our conversation:


And regarding this phrase: "borrowers demand for loans"
I still think that logically makes no sense, and I prefer the first way you stated that: "the borrowers supply of loans"

Substitute "bonds" for "loans" and "bond issuers" for "borrowers" to see why:

"bond issuers demand for bonds"

What? They're the ones that have the bonds to trade!

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