## Wednesday, September 9, 2015

### The emergent representative agent [1]

This is a nice article by unlearning economics on Pieria about some assumptions in microeconomics. I have a couple of things to add to this (I'll leave the great discussion of game theory for another time) ... as well as a possible solution to the issues raised.

First, the assumption of transitivity (or the different but related assumption of GARP) is actually equivalent to the idea that humans have a measure called "utility". Basically all that is involved is that the manifold of [transitive] preferences has a structure that allows them to be related by a diffeomorphism to the manifold of real numbers ... numbers we call utility. A real number we can maximize.

There is no [meaningful] difference between assuming "well-behaved preferences" or "transitivity" and assuming there is a property of economic agents called "utility" measured by a real number.

Another assumption mentioned in the article is monotonicity of utility functions. This is required in order for the utility maximum to saturate the budget constraint and be a single point on that constraint. Again, this assumption is only required because you want to use utility (to come up with a single solution).

So here we are making strange assumptions that are only required because we want to use utility to analyze economic problems.

It reminds me of the history of physics where physicists came up with a bunch of odd assumptions in order to continue to use a theoretical construct. It was called the aether and it asked physicists to assume partial aether dragging and length contraction.

What if I told you you could get the same general results as utility without assuming transitivity or monotonicity? And what if I told you it had a single assumption: the principle of indifference.

Let's start with a basket of consumption goods (or intertemporal consumption periods or both) C₁, C₂, C₃, ... Cn subject to the budget constraint Σ Ci ≤ M.

Let's assume every consumption "state" Ci = pi xi in this d-dimensional space is equally likely (the principle of indifference). This could be intertemporal consumption (i ≤ d indexing time) or different goods (i ≤ d indexing goods) or even intertemporal consumption of different goods (i ≤ d indexing goods and time).

Here is what we have for d = 2:

And here is d = 3:

How can I saturate the budget constraint (and select a point on it) using this? Dimensionality. As d → ∞, the distribution of points becomes highly concentrated around Σ Ci = Σ pi xi = M as can be seen in this graph:

For an infinite number of goods and/or an infinite number of time periods, the 'representative' (average) point approximately saturates the budget constraint. The (emergent) representative agent spends all of its money. However individual agents can vary from spending all to saving all of their money.

This representative agent also appears to engage in consumption smoothing (if you look at i indexing time in the intertemporal problem, all time periods are roughly equal in terms of the value of consumption ... [a symmetry that can be broken by the rate of interest]). Consumption smoothing is an emergent property of the ensemble of agents that are free to choose any point in the domain (and any given agent is unlikely to have very smooth consumption).

This maximum entropy view reproduces the basics of the utility maximization model without the utility. In fact, utility can be seen as emergent [2]. And since utility, a real number, can be used to describe the solutions (equilibria) we see in the maximum entropy view we see that transitivity (or GARP, both equivalent to real number utility) is an emergent property of the emergent representative agent. This is important: transitivity is explicitly not true of the individual agents -- they have random consumption baskets that they have revealed they prefer! Their preferences are not transitive -- they aren't even stable! Agent 9000 prefers A to B one day and B to A another.

So here's a list of some emergent properties of the emergent representative agent:

• Transitive preferences (a consequence of emergent utility)
• Monotonicity of utility (satiation)
• Consumption smoothing

And here's a list of properties of the underlying individual agents:

• Preferences are not transitive and are unstable (random preferences)
• No preference to more or less of a good (random preferences)
• Consumption fluctuates (random consumption)

The idea of a rational utility maximizing representative agent is an emergent construct [1] in the entropy maximization paradigm; real individual people need not have any of these properties.

...

Footnotes:

[1] Noah Smith references emergent representative agents in a recent post:
For example, suppose psychologists find that most human beings are incapable of forming the kind of expectations that time-varying utility models say they do. That would mean one of two things. It could mean that the economy as a whole behaves qualitatively differently than the individuals who make it up (in physics jargon, that would mean that the representative agent is "emergent"). Or it could mean that time-varying utility models must not be the reason for excess volatility.
[2] This is a bit subtle -- entropy maximization chooses a particular utility function (or really a class of utility functions that are maximized at the entropy maximization point).

1. im not so sure why we have to be so fixated on assuming purely random behavior across an unrestricted space.

when we look at the macro variables, we are analyzing whether we lose information or gain information relative to the information equilibrium. but couldn't the perceived number of possible states be increasing or decreasing...thus "correlating" behavior. It seems so strange to me that if the state space is fixed, the interpretation you prefer has people who are seeing a bubble end are losing information...

what am i saying...i don't think you've actually found away out of the individual behavioral realm...

1. another way to pose this same line of questioning might be how do i know the economy is ever out of information equilibrium? what if there really is just less information?

2. The key difference is whether (assuming the model is correct)

p = dD/dS = k D/S

or

p = dD/dS < k D/S

It is true you have to observe the former for some period to see when you've switched to the latter. In the macro model, there is an assumption that outside a recession, you have roughly ideal info transfer. We have some idea of the deviation due to "information loss" in non-ideal transfer vs less information if we look at short term interest rates. There are periods of time where the short rate follows the model and periods where it falls well below the model.

The above post deals with ideal transfer only -- if you have non-ideal, all those atoms (blue dots) will stop being a random cloud and move in the same direction.

3. Another way: It doesn't banish human behavior from a malfunctioning market (non-ideal transfer) -- it just banishes human behavior from functioning (ideal transfer) markets.

4. I think that's why I can't let go of this line of thinking though...at least for business cycle macro, we should care mostly about these potentially malfunctioning markets...or maybe they are efficient and the state space collapsed...

5. I actually agree -- the issue that I have is definition of the "baseline". What does a normal economy look like? What is normal NGDP?

The MaxEnt model gives a specific baseline and defines what the deviations look like (non-ideal info transfer). This picture is different from the more traditional picture of the baseline and recessions look like. For example, David Glasner has a view of coordination and coordination failure -- the MaxEnt view is actually the opposite with random uncoordinated agents in normal times and coordinated (e.g. panicking) agents in recessions.

Nick Rowe sees recessions as a coordination failure brought on by bad monetary policy. The MaxEnt picture says that recessions are non-ideal info transfer that can be independent of monetary policy.

But overall, I agree -- as I put in this picture awhile back:

http://informationtransfereconomics.blogspot.com/2014/01/what-is-and-isnt-explained-by.html

Another way to put it: think of the ITM as theoretically determining the parameters of an HP filter. You don't know what a cycle is until you know what the trend is.

6. I want more than that...I will have to think harder about this loss versus less information...

2. Jason,

Your recent discussions on representative agents (in several posts - not just this one) have left me confused. I asked you several months ago what your sales pitch was for information transfer economics. This is a technique I used as a management consultant. It is a polite way to ask WHAT is it that you are doing (in non-technical terms) and WHY other people would benefit from your thinking. (It is a technique that is very useful when someone focuses mostly on HOW they are doing something and your information transfer techniques are a HOW). However, you avoided the question so I (and probably some other readers) are left to try to figure out our own answers.

My own assumption was that you seemed to be saying that the macro-economy is too complex to be modelled using the arbitrary over-simplifications that are built into standard economic forecasting models e.g. representative agents, specific pricing techniques. Hence, it would be possible to get forecasting results which are just as accurate using a much simpler modelling approach which assumes that almost everything in the macro-economy is random.

I’m still not clear whether that is what you are trying to say or not. However, my confusion arises from the fact that several of your recent posts are about the merits of representative agents. This seems to contradict my original assumption that you were objecting to the artifice of modelling concepts such as representative agents.

If you are now saying that representative agents provide a good way of thinking about the macro-economy then surely that reduces the strength of the argument for moving away from models based on representative agents. After all, there is a whole army of econo-Borg who can produce representative agent models at the drop of a hat, while there is only one of you.

How does this recent representative agent thinking fit into the big picture of your thinking? If my assumption on your sales pitch is wrong then how might I improve it (in a couple of sentences and in plain English)?

1. Hi Jamie,

Your assumption is correct and that is my guiding principle.

And I do think the critique of the representative agent approach is generally valid -- appearing to be a way to get around the SMD theorem; see for example here:

Kirman 1992 [pdf]

However, I also try to make connections with more traditional economic approaches when I can. What I was trying to accomplish here was to show a couple of things:

• The rep agent approach isn't completely flawed
• The traditional way of thinking about the rep agent is flawed
• The resulting MaxEnt agent is very simple

There are two more points that are implicit, but I feel I should have pointed out

• Heterogenous agents aren't the solution to rep agent problems
• The process of adding complexity to rep agents is flawed

The key to that last one is that traditionally rep agents are built as modeling assumptions. I managed to get a few properties common to many rep agent models from randomness. But that doesn't mean all rep agent models (particularly the more complex ones) are possible. In fact, they are likely to be problematic in some way precisely because of the SMD theorem.

It also doesn't mean there are microeconomic implications of macro rep agents. Just because the rep agent does consumption smoothing doesn't mean e.g. you should tax investments differently.

This generally fits with my approach: simpler models (simpler rep agents) may be able to do better than complex ones (complex rep agents).

The traditional economic approach does not use rep agents as simple as the one above and also tends to draw implications for micro agents -- something that I didn't emphasize, but is a flawed approach.

2. Thanks.

3. I noticed that in Unlearning Economics' article (that you link to here) the illustration he used looks kind of like what I was imagining for "the people of the concrete steppes."