The representative agent does not know what he is doing

[This is based on an informal talk I gave to Carleton students. There were first year students, and upper year undergraduates, and graduate students, and faculty, in the audience, which made it a little tricky.]

Somewhere in Canada there is a person whose height is exactly equal (or almost exactly equal) to the average height of all Canadians. I shall call him "George".

George knows his own height (or can measure his own height very easily). But George does not know the average height of all Canadians (he would need to measure the height of all Canadians to know the average height of all Canadians).

George knows his own height. But the Canadian of average height does not know the average height of all Canadians. Because the Canadian of average height does not know that he is the Canadian of average height. And even if he did know that he is the Canadian of average height right now, he would not know if he were still the Canadian of average height tomorrow. Maybe tomorrow it will be Mary.

In terms of height, George is the representative agent. If we multiply George's height by the Canadian population, we get the total height of all Canadians. The representative agent does not know that he is the representative agent.

I like representative agent models. Because representative agent models are a simple way of thinking about microfoundations of macroeconomics. And I like simplicity, and I like microfoundations, despite the problems that come from insisting on microfoundations.

But we need to be careful when we use representative agent models, to avoid fallacies of composition (what is true of each of the parts might not be true of the whole). Each individual knows his own height, and George knows his own height, but does the representative agent know his own height? Well, that depends on what you mean by "his own". Because the representative agent does not know the height of the representative agent. I had to give the representative agent a name -- George -- to explain clearly what I was saying. It is very easy to get confused, and fall unwittingly into a fallacy of composition: "if every individual knows his own height, then everybody knows everybody's height".

(But representative agent models sometimes help us avoid fallacies of composition too: "if every individual can go into net debt to other individuals, then all individuals can go into net debt to other individuals". The representative agent cannot go into net debt to himself.)

Macroeconomics is about (avoiding) fallacies of composition. The fact that each individual firm would increase its quantity demanded if it cut its price does not mean that all firms would increase their quantities demanded if they all cut their prices. Because if one firms cuts its price, that may reduce demand at other firms. Whether it does or doesn't depends on monetary policy.

Macroeconomics is also about coordination problems. Nash equilibria are not always Pareto Optimal; it might be in everyone's interest if each individual did not act in his own interest. The utility-maximising representative agent in a common resource fishery will catch more fish than the number of fish that maximises the utility of the representative agent. And it is not obvious how rational agents even get to a Nash equilibrium. In Nash equilibrium, the representative agent is maximising utility given the actions of the representative agent. But does the representative agent know what he is doing? George knows what he is doing, but George might not know that he is the representative agent.

Take a representative agent macroeconomic model, where the central bank sets a rate of interest, and borrows and lends money freely at that rate of interest. The representative agent first visits the central bank to borrow or lend money, then visits the market to place his orders for others' goods with cash on the nail, and then revisits the market to learn about the orders that others have placed for his own goods.

Does the representative agent know his own stock of money and what orders he has placed for his own goods? That depends on what we mean by "his own". George knows how much money he holds, and how many goods he has ordered. But George might not know that he is the representative agent.

Start in equilibrium, where Md=Ms, and Id=Sd. George wants neither to borrow money nor to lend money. George wants to buy goods for consumption and investment exactly equal to his income from selling goods.

Now the central bank cuts the rate of interest. George decides to borrow money from the central bank, planning to spend more buying goods than his income from selling goods. He plans to invest more than he plans to save. But when George looks at what he has earned from selling goods, he is surprised to discover that he has earned an income from selling goods exactly equal to what he spent on buying goods. He learns that he saved more than he planned to save, borrowed more than he wanted to borrow, and holds more money than he wanted to hold. Id > Sd, and Ms > Md. We are at a point that is off the IS curve and off the LM curve.

And all because the representative agent did not know what the representative agent was doing. George knew what George was doing, but George did not know he was the representative agent.

The representative agent was surprised by his own actions. Will he revise his expectations for next period? Probably, to some extent. But George still does not know why his income was higher than he expected it to be. Maybe it was just a transitory blip in his sales of goods; maybe some of his customers bought goods one week earlier than normal, so that next week his income will be lower than normal.

And when George does revise his expectations, and changes his plans accordingly, that will simply create additional surprises for George, because George does not know he is the representative agent. The representative agent will be again surprised by his own responses to his own past surprises.

And George might also wonder why the central bank cut the rate of interest. Was it because the central bank thought that the representative agent was planning to spend less than his income, at the old rate of interest? Or was it because the central bank wanted to create a boom, by making the representative agent plan to spend more than his income? George does not know.

Will this process eventually converge to a Nash equilibrium where the representative agent knows what the representative agent is doing? George does not know that either.

It might help George figure it out a bit better if the central bank at least told George what level of NGDP it was aiming for. It doesn't solve all of George's problems, but it does make coordination a little bit easier.

Comments

[Nick Edmonds]

I see what you're saying here and, on the whole, I'd agree.

However, it has always been my impression that representative agent models require an assumption that agents are identical (or at least fall into a small number of groups that are identical within themselves). For example, if you assume that households all have a different rate of time preference (even if they had the same basic utility function), I don't think there is any way to represent the aggregate behaviour across time in terms of a single "average" agent, because of the continually changing distribution.

If that's right, then any representative agent model is necessarily about a world of identical agents, so I don't think your concern even comes into play. But maybe I'm wrong on this aggregation point - I've never seen it demonstrated either way.

[Too Much Fed]

"Now the central bank cuts the rate of interest. George decides to borrow money from the central bank, planning to spend more buying goods than his income from selling goods. He plans to invest more than he plans to save. But when George looks at what he has earned from selling goods, he is surprised to discover that he has earned an income from selling goods exactly equal to what he spent on buying goods."

What if George is selling his labor, instead of goods?

What if George decides to borrow to buy an iPhone and Apple saves it?

Does it matter if the central bank buys bonds and then sells the same amount of bonds back to change the rate of interest?

[Nick Rowe]

Nick E: if all agents are identical, and all agents know this, then there is a representative agent, who knows what the representative agent is doing. (But he doesn't necessarily act in the interest of the representative agent).

If agents are heterogenous, it may or may not be possible to construct a representative agent. For example, the Condorcet paradox, with three policies A B and C, where one agents ranks policies ABC, the second agent ranks them BCA, and the third agent ranks them CAB. With majority voting, the representative agent has intransitive preferences. But if all agents have single-peaked preferences, we can (I think) construct a representative agent. It depends.

(And for some questions, of course, a representative agent model won't work well at all. Like if we want to understand the distribution of net debt.)

TMF: you are wandering off-topic, and totally ignoring the point of the post. Stop trolling by asking red-herring questions.

[Oliver]

Still trying to wrap my head around this one.

My first thought was, the representative agent not only does not know what he's doing, she also has multiple personality disorders...

Other random thoughts:

1.Because if one firms cuts its price, that may reduce demand at other firms. Whether it does or doesn't depends on monetary policy. I don't get that.

2.George decides to borrow money from the central bank, planning to spend more buying goods than his income from selling goods. He plans to invest more than he plans to save.

George has an income of $1'000 of which he needs every penny to pay for his expenses. So he usually saves $0 per month.

This time round he borrows $5'000 with which he buys a new machine. So this period, he saves $1'000 - $1'000 - $5'000 (loan) + $5'000 (machine) = $0. Same for as it ever was for George in terms of net worth. So, before his machine starts producing magical widgets, none of these three statements seem true for George:

He learns that he saved more than he planned to save, borrowed more than he wanted to borrow, and holds more money than he wanted to hold.

But what is true is that there are now $5'000 as well as a machine more in the economy than before. And that those $5'000 are now ready to flow back to George as income for widgets.

I still don't understand why either George or the representative agent should be surprised about this, though. It's precisely what they wanted. They wanted more and that's exactly what they got.

What they don't know a. why and b: is how it'll turn out, which might also be influenced by interest rate movements, gvt. action, mass panic, foreigners or earth quakes.

I'm sorry if my learning is flatter than a Post-Keynesian LM curve...

[Frank Restly]

Nick,

"It might help George figure it out a bit better if the central bank at least told George what level of NGDP it was aiming for."

I think that George would soon figure out that whenever NGDP did not hit the central bank's target, the central bank would cut the interest rate offered to him.

George would revise his expectations of both people buying goods from him AND his expectations of the central bank's reaction function. Suppose NGDP was below target - presumably in this model, the central bank lowers the interest rate it offers to bring NGDP back to trend. Why shouldn't George wait until the central bank is offering money at 0% interest before acquiescing to a loan?

George will figure it out all right. He will figure out that he has the central bank by the short hairs.

[Nick Rowe]

Frank: fallacy of decomposition. What is in the interests of all is not necessarily what is in the interest of each.

Stop now.

[Miami Vice]

why doesn't George know why the central bank lowered interest rates? is that an assumption explicit in the model or is it a fact that lowering/raising interest rates is ambiguous as policy response?

[Nick Rowe]

Miami: well, maybe the central bank thought that the representative agent was planning to spend less than his income, so it needed to cut rates to prevent that happening, and prevent inflation falling below target. And if George thinks the central bank is right about that, George would not expect his income to rise when he sees the central bank cut the interest rate. But if George thinks the central bank cut interest rates to try to create a boom, he would expect his income to rise. (Assuming George has rational expectations, and can figure this all out).

[Miami Vice]

the expectation that the central bank hits its target is the reason that changes to interest rates by the cb don't affect expectations. is that right?

[Gene Callahan]

Great post Nick! The kind of thing that makes me keep checking in here.

[Gene Callahan]

Frank: "Why shouldn't George wait until the central bank is offering money at 0% interest before acquiescing to a loan?"

And why don't consumers just not buy TVs until the TV manufacturers simply give them away to stop paying to warehouse them? Because if consumers were acting in unison, this could be achieved. But, in fact, at some point individuals say "That's cheap enough for me!" and they buy. The representative agents is not some Rousseauian "general will" of all agents!

[Oliver]

Right, second attempt (actually third, I ditched No.2)

Can I summarise your views as follows.

Individual actions have consequences that lie beyond the individual's control.

In some cases, individual actions may lead to offsetting actions by the other individuals, leaving the average (representative agent) unchanged.

In other cases, individual actions will not lead to offsetting actions, thus changing the average.

The actor cannot be sure which of either outcomes will ensue. To the extent that the outcome differs from his/her expectations he/she will be surprised.

With sufficient data, one can watch the average and attempt to move it towards a designated policy goal with a couple of tools known as monetary and fiscal policy. One can communicate that policy goal in advance in the hopes that the average will change less erratically as a consequence.

[AlanDownunder]

If George is a rational intertemporal optimiser, he is remarkably unrepresentative.

[Nick Rowe]

Miami: "the expectation that the central bank hits its [NGDP] target is the reason that changes to interest rates by the cb don't affect expectations [of NGDP] . is that right?"

Yes, with my changes added. But George's expectation of changes in NGDP are not necessarily the same as George's expectation of changes in George's nominal income.

Let me put it this way: when the central bank cuts r, that might or might not affect George's expectation of George's income. It depends, on a lot of things. But George does not know that George's income will be exactly equal to George's expenditure. Even though it is true. Because George does not know that he is the representative agent.

Gene: thanks! (I think I'm channeling Hayek. Hayek's the man, for this sort of stuff. If only he had written more clearly and simply.)

Oliver: try it this way: each individual's actions depend on his expectation of the actions of others. (His expenditure on others' goods depends on his expectation of others' expenditure on his goods.) But he does not know what others will do, and he does not observe their actions until after he has chosen his own actions. The representative agent always earns an income exactly equal to his expenditure. So if he knew he was the representative agent, and he knows his own expenditure, he would also know his own income. But he does not know he is the representative agent.

Alan: I made no assumption here of intertemporal optimisation. I assumed only that expenditure depends on income and the rate of interest. Just like Keynes.

[Bob Murphy]

Oh come on Nick, you had such a great post going until the very end. I don't know who this guy George is, but I'm quite sure he doesn't give a damn about NGDP.

[Nick Rowe]

Bob: OK, fair enough. That last little bit does need to be fleshed out some more. (You Austrians should read the expurgated version, minus those last two lines!)

George wants to know George's (nominal) income. Knowing NGDP (divided by population) will tell him the nominal income of the representative agent, and how it is changing over time. That is one less thing he has to figure out. Now he can concentrate on his comparative advantage -- figuring out what makes demand for George's goods special -- rather than trying to figure out macroeconomics too.

[Miami Vice]

Under what circumstances is it possible get off the circle of self fulfilling expectations?

[Min]

"George wants to know George's (nominal) income. Knowing NGDP (divided by population) will tell him the nominal income of the representative agent, and how it is changing over time."

If George knows that NGDP/pop = his nominal income, then he knows that he is a representative agent. ;)

[Min]

The representative agent is a simplifying assumption. That the representative agent does not know that he is a representative agent is another simplifying assumption. That the representative agent does not act as though he knows that he is a representative agent is another simplifying assumption.

When and to what degrees these assumptions are true or useful are interesting questions. :)

[Min]

The representative agent is a simplifying assumption. In this day of vastly powerful computer simulations, why bother with such simplifications? After all, they can hardly be representative of any complex economy.

[Too Much Fed]

Nick, I actually read the post too quickly.

Does the central bank just announce what the rate is? It does nothing else tangible (announcing a NGDP target is not tangible)?

[William Polley]

Preferences that are of the Gorman form are necessary and sufficient to ensure that heterogeneous agents can be represented by a representative agent. (See, for example, Hal Varian's graduate micro text.)

That being said, the usual assumption in macro (at least for starters) is to assume identical agents. Many representative agent models begin with a line such as "...a continuum of homogeneous agents with unit mass..." or something along those lines.

So does the representative agent know he's the representative agent? If the agents are identical and there is perfect information, I don't see how he doesn't know. He must know (as your first comment, Nick R., acknowledges). But then I'd counter with the following question. If he's a price-taker, does it matter? Hence the usual assumption of a unit mass of agents with any single agent having measure zero. No agent can affect the price. If he can't affect the price, then there's no reason to deviate from the symmetric equilibrium. He may very well know that he has the same characteristics as everyone else, and therefore is identical to "the" representative agent. However, since he cannot exploit that knowledge, it doesn't really matter.

So in other words, the representative agent just maximizes his own utility given prices (which are determined by everyone else's choices that he can't control). His utility maximizing choice (identical to that of everyone else) may not maximize social utility. Your example of the common fishery is a good one. But this is not a result of his not knowing he is the representative agent (as your original post suggests); it is a result of his not being able to control the actions of others. The competitive equilibrium is not Pareto optimal because the representative agent is not a social planner. He has different constraints. This is not an inconsistency of the representative agent model. It is exactly how it is supposed to work.

[William Polley]

Shorter version: There are plenty of problems with the representative agent paradigm. This isn't one of them.

[Nick Rowe]

William: thanks for the useful comment.

But I don't see this as a *problem* with the representative agent paradigm. I see this as a way to *extend* the representative agent paradigm. Keep the representative agent; drop the assumption that the representative agent knows he is the representative agent. Add a mix of aggregate and agent-specific shocks, where each agent observes only the sum of the two shocks, and you have a model where the representative agent thinks he is a special snowflake.

I had a Bertrand model at the back of my mind.

" Your example of the common fishery is a good one. But this is not a result of his not knowing he is the representative agent (as your original post suggests); it is a result of his not being able to control the actions of others."

Agreed. My original post maybe wasn't clear enough. Those are two separate issues. Even with identical agents, where each knows he is the representative agent, Nash Equilibrium may not be welfare-maximising, as is well known. I wasn't saying anything new in that bit.

[Bob Murphy]

Nick you provoked me to do a whole post on this. Just skim the second half, you'll see where I took it.

[Nick Rowe]

Thanks Bob! I left a comment there.

[Jeff]

To me the key point is this one: "And even if he did know that he is the Canadian of average height right now, he would not know if he were still the Canadian of average height tomorrow. Maybe tomorrow it will be Mary." The RA is a reification of something that is not real/stable -- there is no "the" representative agent, and therefore there is no planning/decision-making/agency by a RA. The term is oxymoronic. Maybe that is (part of) your point?

Like the "center of mass" of an object in physics, you could say that there is a "central mass molecule" at any given time, the molecule that happens to be nearest the theoretical location. But any movement of that molecule immediately causes it to lose that status.

Interesting argument though as to ignorance in individual economic decision-making. It could explain how masters of the economic universe (those with more influence on policy and/or those with inside information on same) manage to exploit the average joe without his knowledge. "Hey, I thought I was doing OK, keeping up with the Joneses, and all, but whoa, what did you say CEOs now are pulling in again??"

[Nick Rowe]

Jeff: "The RA is a reification of something that is not real/stable -- there is no "the" representative agent, and therefore there is no planning/decision-making/agency by a RA."

That is true, though it wasn't my (main) point. Take an example of a forest with an infinite number of trees, with the same number of trees being born every year. Each tree grows at exactly the same rate, and then dies and falls over when it gets to 100'. The "representative tree" is exactly 50' tall, and is not growing at all.

[Jeff]

@Nick, thanks.

I understand, "the" RA is a mathematical construction. My concern/objection is I guess on two related points, 1) the rhetoric/language/jargon attempting to explain the concept is extremely fraught -- there is no "the", there is no individual in the population (use of "George" is misleading), and there is no agency involved (the term itself is a misnomer); and 2) especially so when the media and the lay population get ahold of it and go to town.

[Nick Rowe]

Jeff: well, a representative agent model *does* ascribe agency to the representative agent. The representative agent chooses actions, given his expectations, and maximises utility. And the idea is that by understanding the actions of the representative agent, you can understand the economy. But, as you say, this is problematic if the representative agent is changing over time. Right now it's George, but next period it's Mary. So if George invests this period, Mary next period does not own George's investment from last period.

[Oliver]

The representative agent always earns an income exactly equal to his expenditure. So if he knew he was the representative agent, and he knows his own expenditure, he would also know his own income. But he does not know he is the representative agent.

What is this agent a representative of, exactly? Surely, the answer should be 'reality'.

So, one can take the economy as a whole, or some aggregate measure of it, and then divide that by the number of people within the economy to create a synthetic creature that one can call representative agent.

I can't say whether it matter if this creature is given a mind of his own, knows who he is, behaves in unisone with his compatriots or wheter he is omniscient, or not.

But surely, what does matter, is whether we can take this creature, multiply his actions / decisions by the number of people in the economy and arrive at the status quo. Because if we can't do that, say by limiting his decisions to conform with some outside view of rationality, e.g. by saying he spends all his income (=no saving), then he is no longer representative of reality. He is not representative of what is, but of what someone believes he should be . If that isn't a fallacy of composition, I don't know what is.

[Jeff]

@Nick, thanks for the further explanation.

Yes, you are getting at my critique. Even further, I see you are talking about a discrete-time model with a period of something like a day. How is anything based on that going to fare empirically when markets today are computer-traded at millisecond periods? Won't any policy based on it be arbitraged into oblivion? Or does the necessary human scale of the economic decision-making (i.e. changes) save it? Apologies for my naïveté.

[Jeff]

@ Oliver re: status quo. The model is by definition a representation of "reality" (caveat: as close as can be measured). And in a closed macroeconomy there are no incomes which are not also expenditures (by someone else). Or to put it prosaically, quantitatively there are just as many people who spend more than they make as there are people who make more than they spend. "The representative agent" is therefore someone exactly in the middle, spending exactly the amount it makes.

[Nick Rowe]

Jeff: good question re continuous time. (But I wonder if there isn't some variant on the fallacy of composition as we move towards continuous time; even if each individual can adjust instantly might not mean that the economy as a whole can adjust instantly). The way I think of it: even though time is continuous, flows of information are not instantaneous, and information flows are lumpy. If you are selling cars, you do not sell 0.001 cars per minute. If you normally sell 1 car per day on average, you might not revise your expectations much if you sell 2 cars some days and 0 cars other days. It takes time for you to learn whether your demand curve has shifted, if there are permanent and transitory shocks.

[Oliver]

@ Jeff

That's true for a closed system. I thought Nick had a world in mind in which the monetary system is ex machina, i.e. in which money creation does not count towards the representative agent's expenditure.

[Jeff]

@Nick Yes interesting point re chunkiness. There is a lot of variability even there though. On the high side, you've got Trump buying buildings and M&A. On the other side you've got your amazons and McDonaldses that pretty much have an instantaneous read on sales. (And the aforementioned financial markets.)

[Oliver]

... And if Nick has a closed system in mind, then that answers his question from a previous post as to whether money constitutes net wealth. In that case, no, it doesn't.

[Too Much Fed]

"George decides to borrow money from the central bank, planning to spend more buying goods than his income from selling goods."

Is there a labor market in this model? I'd say selling labor and selling goods are different.

"But when George looks at what he has earned from selling goods, he is surprised to discover that he has earned an income from selling goods exactly equal to what he spent on buying goods."

If I'm reading that right, George borrows $20,000 to buy a $20,000 car. You are saying the $20,000 comes back to George?

Here is what happens in the real world. Let's say George sells fishing lures. George borrows $20,000 to buy a $20,000 car so the $20,000 enters the goods/services economy. Somewhere along the line another entity removes the $20,000 from the goods/services economy. It holds the $20,000.

George ends up with the same income from selling fishing lures.

[Min]

Too Much Fed: "If I'm reading that right, George borrows $20,000 to buy a $20,000 car. You are saying the $20,000 comes back to George?

"Here is what happens in the real world. Let's say George sells fishing lures. George borrows $20,000 to buy a $20,000 car so the $20,000 enters the goods/services economy. Somewhere along the line another entity removes the $20,000 from the goods/services economy. It holds the $20,000.

"George ends up with the same income from selling fishing lures."

In that case, George is not a representative agent, in the intended sense. As a representative agent, George is Everyman. That means that the other entity is also George, or a collection of Georges. So while some George borrows and spends the money, but does not get an increase in income, that is not true of every George. If each George borrows and spends $20,000, then the average income of all the Georges increases by $20,000. The ones whose income stays the same are unlucky.

There is another sense of representative, expressed in the phrase, average Canadian, who indeed might borrow and spend $20,000 and yet see an increased income of, say, only $15,000. The extra $5,000 being sucked up by the rentiers.

[Min]

Nick Rowe: "Take an example of a forest with an infinite number of trees, with the same number of trees being born every year. Each tree grows at exactly the same rate, and then dies and falls over when it gets to 100'. The "representative tree" is exactly 50' tall, and is not growing at all."

This indicates a problem with the idea of microfoundations. If we look at the behavior of each tree (if we allow ourselves to talk about tree behavior :)), then each tree is growing. The only reason we know that the "representative tree" is "not growing", in the sense that the average height of the trees remains the same, is precisely by ignoring microfoundations and looking at macro statistics. Then we are deriving our so-called microfoundations from macro considerations.

[Too Much Fed]

Nick said: "Start in equilibrium, where Md=Ms, and Id=Sd. George wants neither to borrow money nor to lend money. George wants to buy goods for consumption and investment exactly equal to his income from selling goods.

Now the central bank cuts the rate of interest. George decides to borrow money from the central bank, planning to spend more buying goods than his income from selling goods. He plans to invest more than he plans to save. But when George looks at what he has earned from selling goods, he is surprised to discover that he has earned an income from selling goods exactly equal to what he spent on buying goods. He learns that he saved more than he planned to save, borrowed more than he wanted to borrow, and holds more money than he wanted to hold. Id > Sd, and Ms > Md. We are at a point that is off the IS curve and off the LM curve."

Min said: "If each George borrows and spends $20,000, ..."

Min, let's say everybody is like George at the beginning (George wants neither to borrow money nor to lend money. George wants to buy goods for consumption and investment exactly equal to his income from selling goods.)

Now cut the "fed funds rate" with the assumption that the central bank is the "only bank" (an unrealistic assumption).

George borrows $20,000 and spends it. Every other George does what every other George did before.

What happens? Is there any representative agent here?

[Too Much Fed]

I said: "Does it matter if the central bank buys bonds and then sells the same amount of bonds back to change the rate of interest?"

Disregard that. I read the post too quickly. It appears there are no commercial banks here.

[Nick Rowe]

TMF: stop commenting please. I don't think you understand the post at all.

[Min]

Too Much Fed: "George borrows $20,000 and spends it. Every other George does what every other George did before.

"What happens? Is there any representative agent here?"

It looks like you have defined one away.