Two LM curves

Sometimes we borrow money from the bank because we plan to spend more than we expect to get in income. And sometimes we borrow money from the bank because our stock of money is too small relative to our flows of planned spending and expected income.

Here is a rough sketch of a simple model that captures that idea. It is an ISLM model, except it has two LM curves. One LM curve is horizontal ("Keynesian"), and the other LM curve is vertical ("monetarist"). You can only ignore the vertical "monetarist" LM curve if you are willing to assume fully rational expectations and full information, so each individual agent knows what every other agent is planning to do, and can solve the model for the equilibrium level of output, which he knows with certainty.

At the beginning of the period, all agents announce the prices at which they will sell goods. Prices are fixed in advance. Output is demand-determined at those fixed prices.

Next, the bank opens, announces a rate of interest, and offers to lend or borrow money at that same rate of interest. Agents choose how much money they will hold given that rate of interest. The bank than closes.

Next, Nature tosses a coin for each agent, which determines the order in which they go shopping. There is a very large number of agents, so exactly half the agents are early shoppers, who will spend money before they earn it. And half the agents are late shoppers, who will earn money before they spend it. The early shoppers buy from the late shoppers, then the late shoppers buy from the early shoppers.

The early shoppers will want to ensure their stock of money is sufficient to satisfy their cash-in-advance constraint. The late shoppers will want a zero stock of money, unless they plan to spend more than they expect to earn. But agents do not know whether they will be early shoppers or late shoppers until after the bank closes. They choose a stock of money which depends on how much they plan to spend, on how much income they expect to earn, and on the rate of interest (because that is the opportunity cost of having unspent balances at the end of the period).

What would the aggregate demand function look like in this economy? There are two different ways we could write the AD function:

1. Yd = F(Ye,r). Output demanded Yd is an increasing function of expected income Ye, and a decreasing function of the rate of interest r set by the bank. This is how we would write the demand function before the bank has closed its doors and before we know M/P.

2. Yd = H(M/P). The early shoppers spend all their money (their cash-in advance constraint will be binding, if r>0), which determines the income of the late shoppers, and how much they will spend. This is how we would write the demand function after the bank has closed its doors and after we know M/P.

(Both demand functions will also depend on expected income and expected interest rates and expected prices in all future periods, but I have ignored those expectations of future periods here.)

With rational expectations, and identical agents (except for Nature's coin toss to determine who shops when), each agent could solve for Ye=Y=Yd from the first AD function, given the rate of interest set by the bank. Solving the first AD function gives us a standard IS curve, with Y a decreasing function of r. That IS curve, plus the horizontal LM curve, with the bank setting r, lets us solve the model. The second AD function isn't needed, unless we want to solve for M/P.

But suppose agents cannot solve the model to learn Y with certainty, because Nature also adds some aggregate shocks and agent-specific shocks, and individual agents cannot separate the two, and so do not know how much other agents are planning to spend, and so do not know how much income the late shoppers will earn from the early shoppers? Then the second aggregate demand function will matter too. The late shoppers will find that their expectations of income from the early shoppers may not be realised, and will revise their planned expenditures. If the early agents actually hold less money than the late agents expected them to hold, and so spend less than the late agents expected them to spend, some late agents may find their cash-in-advance constraints binding, and all late agents will buy less than they had planned to buy, because they have learned that their incomes are lower than they had expected them to be.

Under rational expectations and full information (except for the coin toss) we can ignore the second AD function. We set Ye=Yd=Y, solve the first AD function for the standard IS curve, add a horizontal LM curve for r, and those two curves determine Y.

Otherwise, if we take expected Y as given, M/P will be a function of Ye and r, and we need the second AD function to determine Yd and Y given M/P. Output is determined by the vertical LM curve.

The standard IS curve, which assumes Ye=Yd=Y, is an ex-ante IS curve. It tells us what agents would rationally expect the level of output to be, given the horizontal LM curve. And those two curves determine the rationally expected stock of money, and where agents rationally expect the vertical LM curve to be. Agents with model-consistent expectations expect all three curves to cross at the same point.

But if the representative agent has false expectations, because he does not know he is the representative agent, and so plans to spend more or less than he expects to earn, the actual vertical LM curve will not be where he expects it to be, because agents actually hold either more money or less money than he expected them to hold. And it is that actual vertical LM curve that determines actual output demanded and actual output. And this period's actual output will in turn affect expectations of future income in future periods.

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

Comments

[Tom Brown]

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.

[JW Mason]

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."

[Nick Rowe]

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.

[HJC]

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.]

[Nick Rowe]

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.

[Nick Rowe]

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.

[Maurice Lechat]

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?

[Nick Rowe]

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.

[Squeeky Wheel]

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).

[Nick Rowe]

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.

[Nick Rowe]

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.

[Odie]

"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.

[Maurice Lechat]

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?

[Tom Brown]

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

http://www.themoneyillusion.com/?p=26400&cpage=1#comment-324718

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!