Emergent representative agents: a means to an end

David Glasner responds to Tom Brown:

The problem with an emergent representative agent is that you need to explain emergence before you know what the emergent representative agent actually looks like, so I don’t see that arguing in terms of an emergent representative agent actually accomplishes anything. It still seems to me like a form of question begging.

I'd completely agree with David's sentiment in the case where you are actually trying to understand how something works. That is to say my recent set of posts talking about an emergent representative agent are actually meta-arguments. Updating my language a bit to use Gary Becker's paper, I am trying to argue three things:

• The use of rational agents is not an immediate reason for mainstream economic theory to be wrong on its face. There's a lot of what Noah Smith calls lazy econ criticism that makes arguments like 'economists assume rational people, but people aren't rational ... LOL'. If a seemingly rational agent on average can emerge from irrational behavior, then takes the wind out of the sails of behavioral theories.
• Microfoundations are probably irrelevant to macroeconomics. Macroeconomics can have very different properties from its microfoundations. This should have been well understood after the SMD theorem, but as Kirman [pdf] says the representative agent tries to sneak around it. Showing that the emergent representative agent relevant to macro can have very different properties from the individual micro-agents should put a stop to the sneaking.
• Since rational agents can emerge from solely the properties of opportunity sets, let's skip the middleman (middle-agent?) and just use the mathematics of opportunity sets. The mathematics of opportunity sets is information theory (what messages could be constructed "given the opportunity" to use x bits). This information theory leads to basic supply and demand logic (but allows for specific failure modes) and is the impetus behind my paper.

Only the third one would be directed at David Glasner since he agrees with the second and doesn't make the first argument.

The first argument amounts to a defense of a swath of traditional economics. The second is a swipe at a different swath. The third is a potential third way -- or more like a rethink of the traditional diagram approaches.

When is an intertemporal budget constraint a true budget constraint?

David Glasner cautioned me about the use of an intertemporal budget constraint (since it is based on expectations that could be thwarted) in my emergent representative agent argument (that parallels Gary Becker's argument that even irrational agents can behave rationally). If you have no idea what I am talking about, you should start here. At that link, I said I would address this issue in a future post -- this is that post.

The question is whether we can pretend the intertemporal budget constraint is a true budget constraint analogous to a single period budget constraint for the purpose of constraining the (intertemporal) opportunity set (as Becker puts it). I called the opportunity set the simplex. The issue is that recessions happen and output falls below its expected value (plans are thwarted). That means that expected value isn't a true constraint (actual output is much less) and therefore the most likely state of the economy might not:

• Manifest consumption smoothing
• Saturate the intertemporal budget constraint
• Have downward sloping demand curves
• (Approximately) maximize utility

The crux of the argument showing these hold (if we ignore recessions) with random consumption is that the most likely point of a high-dimensional simplex is on its surface, rather than in the interior. The downward sloping demand curves hold as long as the centroid of the simplex isn't close to any particular axis.

Let's take the opportunity set defined by the intertemporal budget constraint Σ ci ≤ M (see the picture at the top of this post). For d dimensions (time periods) we have

Σ ci ~ M d/(d + 1)

If d >> 1, then we have Σ ci ~ M and the above properties hold: we have emergent rational agents.

What happens if we introduce recessions? Let's take a set of {ci} of size n to be zero: all plans are thwarted in those periods and there is no income ... a 100% recession. Note that for even a bad recession, you're really looking at 20%, not 100%, so this will be a conservative calculation. We can show

Σ ci ~ M (d - n)/(d + n + 1)

However, if d >> n and d >> 1, we still recover Σ ci ~ M. In fact, we still recover all of the properties listed above (if d >> n, then the centroid isn't close to a typical axis and you have downward sloping demand curves almost always). Basically, as long as the economy isn't in a recession most of the time, the intertemporal budget constraint can be treated as an ordinary budget constraint.

Note that the properties hold even if the consumption sets are of different size (i.e. there is economic growth) as long as there is no cj >> ck for most k (in which case Σ ci ~ cj). That case is a lopsided simplex (the opportunity set would be more like a spike along the dimension j).  Can growth make our simplex lopsided? The Euler equation says that successive consumption periods are related by the rate of interest  and we have:

ci ~ β (1 + R) cj

This means that the rate of interest can't be too large. Large here though is a bit extraordinary; we'd need

β c0 (1 + R)^d >> c0 (1 + R)^(d - 1)

or

β (1 + R) >> 1

to make a truly lopsided simplex.

So the final takeaway is that we can treat the intertemporal budget constraint like a true budget constraint in order to demonstrate emergent rationality from random behavior. The caveats are:

• The economy is not in a recession in most time periods
• Consumption isn't concentrated in a few time periods

It is true that when you look at time periods near recessions, things are more interesting. However that is the point of non-ideal information transfer. In the information equilibrium model, information equilibrium is taken to hold most (but only most) of the time.

Core CPI and lags

The latest core CPI data came out today and is up on FRED. I thought I'd post a picture of the model results without smoothing as I did for the PCE numbers that came out at the beginning of the month. However I noticed a pretty obvious lag in the CPI data relative to the model (there is no obvious lag in the PCE inflation data [1]), so I decided to try and do a four-parameter fit with a lag y0 (along with α, m0 and γ -- see paper).

Overall, it works pretty well (model = blue, data = green and 1-sigma error bands) with the lag being y0 = 1.2 years:

And here's a zoom to the last 15 years:

If the lag is correct, then the model gives the future values of core CPI (past the vertical line) using data available today.

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Update

Although less obvious, the lag has the opposite sign for Japan (y0 = -0.9):

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Footnotes:

[1] Here is the graph for reference:

One interesting thing is that maybe PCE inflation measures CPI inflation y0 = 1.2 years in the future -- which might (partially) explain why PCE inflation comes in about 0.3 percentage points lower (per Scott Sumner).

Potentially file under: thinking we know more than we do (corrected)

Correction 10/29/2015: GDPnow forecasts real GDP, not nominal GDP. I completely missed that "real" in the graph label, and on the website ...  so you can pretty much ignore the post below. The update does actually notice that it is real GDP.

Original post:

I thought readers might be interested in (and entertained by) the rather vague ITM forecast of NGDP for Q3 of 2015. It appears as the orange line above (the forecast has not changed since July of this year, so it is a horizontal line). The orange bands represent the 50% and 90% errors (roughly 0.6 and 1.6 sigma) based on data from 2010 Q1 - 2015 Q2.

The key take-away is that ITM sees most of the fluctuations of NGDP as random noise without seasonality -- it's supposed to be SAAR, anyway, right? A really high result could be seen as 'vindication' -- really just vindication of randomness. I'll probably just find it funny if the Q3 result does come in fairly high. The ITM doesn't really give you much more information than a log linear fit.

The GDPnow value from the Atlanta Fed may well be better (the source of the diagram above).

Does anyone know the 2015 Q3 prediction from hypermind? The site only has the 2015 annual level, which has gone down to 3.2% despite some upward revisions in previous quarters making me think it is giving a result closer to the blue chip or GDPnow value.

Update 10/19/2015:

Here's some results from Macro Advisors (H/T Brad DeLong) for real GDP:

Gary Becker's emergent rational agents

The second graph is my maximum entropy version of the Diamond-Dybvig model, but NR and B correspond to p' and p in Figure 2 from Gary Becker's paper below.

David Glasner identified the argument I was making here (and in the links here) as one made by Gary Becker in 1962; here's the comment from David:

Jason, Thanks for the link and your follow up post as well. Your argument reminds of the paper "Irrational Behavior and Economic Theory" published in JPE in 1962 and reprinted in Becker's The Economic Approach to Human Behavior in which he showed that budget constraints were sufficient to imply negatively sloped demand curves and other standard microeconomic results. He credited Alchian's 1950 paper in JPE "Uncertainty, Evolution, and Economic Theory" for anticipating his argument. As I recall, Israel Kirzner wrote a comment published by JPE criticizing Becker for not sticking with utility maximization. It might help you to use Becker’s argument as a way of improving your communication with economists who, if they are like me, have trouble comprehending arguments that aren't made in the language we're accustomed to speaking. One other point to consider is that in an intertemporal context with incomplete markets there is no such thing as a true budget constraint because the prices in the budget constraint are largely expected prices not actual prices, so if prices turn out to differ from those that were expected, budget constraints may be violated (households or firms go bankrupt).

Regarding the language, I completely agree that is an issue. Regarding the intertemporal budget constraint, a post is forthcoming (though it basically involves combining these three posts [1], [2], [3]).

Speaking of language, language was the reason it took me a little while to parse Becker's version of the argument (I have a hard enough time parsing even physics papers [pdf] from the 1960s), but it is pretty much identical to ones I've been making. There are three key differences as I mentioned in a reply to David on his blog and thought I would expand here.

1) [I give] an argument that if the number of different goods [or intertemporal periods] d being optimized (the number of axes) is large, there is no need to restrict to the budget constraint (as ‘maximization’ happens automatically as d → ∞)

As Becker writes:

If opportunities were initially restricted to the budget line AB in Figure 2 [reprinted above], the average consumption of many households would be close to p, the midpoint of AB, with different households uniformly distributed around p.

And later:

Inefficient impulsive households might assign equal probabilities to all points in the opportunity set, not just to those on the boundary. The average consumption of a large number of these households would almost certainly be at the set's center of gravity, with households uniformly distributed around this point. ... For example, point c in Figure 2 [reprinted above] would be the center of OAB and c', to the left and above c, would be the center of OCD.

The existence of a large number of goods (or intertemporal periods) d leads to the average being near p without restricting to the line AB. As d → ∞, we have c  → p because most of the points in a high dimensional volume are near the surface, not the interior.

2) [I present] an interpretation of the equilibrium in terms of entropy

When Becker says uniformly distributed (and equal probabilities) in the quotes above, he is making a maximum entropy (least informative prior distribution) argument. Therefore, information equilibrium relationships will apply.

3) [There is] the possibility of falling away from the ‘maximum’.

A simple random walk around the simplex leads to cases where occasionally the budget constraint isn't saturated as illustrated (for a single period and d >> 1 goods markets) at this link.

Overall, Becker's paper is a pretty cool common point of reference.

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Update:

I would like to emphasize that while Becker refers to agent actions as irrational (in the title) or "Inefficient" or "impulsive", I take the view that those actions may just be more complex than we can model efficiently ... for example, so complex as to appear random from an outside viewer.

Update, the second:

This Socratic dialog could be seen as Gary Becker arguing with Kirzner ...

The representative macro-theory agent differs from micro-theory agents

I wish I could write something clear enough because I think once I do David Glasner would champion the maximum entropy approach. Last night he wrote about representative agents having different properties from the micro agents. He used a traffic model as an analogy:

Consider a traffic-flow model explaining how congestion affects vehicle speed and the flow of traffic. It seems obvious that traffic congestion is caused by interactions between the different vehicles traversing a thoroughfare, just as it seems obvious that market exchange arises as the result of interactions between the different agents seeking to advance their own interests. OK, can you imagine building a useful traffic-flow model based on solving for the optimal plan of a representative vehicle?

No, of course not. The 'representative vehicle' (that travels at some varying velocity, carrying a varying number of passengers) is an emergent degree of freedom that simplifies the theory.

In physics we have quasi-particles (collective excitations) like phonons. In fact, you'd probably describe a traffic model as a linear combination of vehicle phonons:

The atoms in a lattice don't travel anywhere, yet phonons (emergent waves that don't exist for individual atoms) carry the energy of a shock to the lattice across it. The properties of a material like its head heat capacity and temperature follow from maximum entropy distributions of phonons.

In the information transfer traffic model, the underlying micro-theory vehicles have random velocities, yet the emergent 'representative vehicle' has a single [stochastic] velocity. It would be a mistake to associate the properties of the representative vehicle with the micro-theory vehicles.

In this post, I construct a representative agent from a collection of micro-agents with completely different properties. Micro agents do not have transitive preferences, do not maximize consumption nor do any consumption smoothing; the representative agent has all of these properties (even a well-defined utility function).

Reading Glasner's post gave me the feeling that my blog is like screaming through sound proof glass. Here's Glasner just before the previous quote:

... what I was trying to argue was ... that representative-agent models suffer from an inherent, and, in my view, fatal, flaw: they can’t explain any real macroeconomic phenomenon, because a macroeconomic phenomenon has to encompass something more than the decision of a single agent, even an omniscient central planner.

I think Glasner means the representative agent solution can't explain any real macro phenomenon in terms of micro agents [1]. And that is true of the emergent representative agent. The micro agents and the representative agent have little in common (except maybe the properties allowed by the SMD theorem).

There is a separation between the micro agents and the macro properties and when this separation occurs, the macro properties (given by the representative agent) are the result of the bulk properties of the economic state space. Sometimes this separation doesn't occur (or breaks down), and the representative agent dissolves into a complicated simulation with millions of agents. This is the main point of my earlier post and the idea there is summarized in this graphic:

This happens in disordered systems (like glasses) as well as so-called meso-scale physics where the phonons become less well-defined. In that case there is no simplification, and you have to treat the problem as made up of millions of individual atoms [2].

The key thing to understand here is that the macro theory may have little or nothing to do with the micro theory. The theory of quarks has little to do with the theory of protons and neutrons (except some bulk properties). The theory of phonons (lattice excitations) has little to do with the theory describing an individual atom. The theory of an ideal gas has little to do with the theory of individual atoms. Actually, the emergent macro-theory tends to have more to do with just the symmetries and bulk properties of the state space rather than the details of the micro-theory.

In the quark case it's actually pretty interesting -- there is no scale at which both the quark and hadron theories are simple descriptions. At high energy, the quark theory simplifies. At low energy, the hadron theory simplifies [3]. In physics we call this duality -- sometimes the wave description of a quantum system simplifies and sometimes the particle description simplifies. Some phenomena are more easily seen as electric fields and moving charges, some phenomena are more easily seen in terms of magnetic fields.

For economic phenomena, sometimes the representative agent simplifies and sometimes micro-theory agents simplify.

Economics doesn't have a good way to tell which is applicable when (scope conditions) yet, but the information equilibrium model is a good start, IMHO.

Footnotes:

[1] If Glasner doesn't mean this and means that even an emergent representative agent can't be used, then I'd say we have disagreement. But we'd agree again if what he means is that an emergent representative agent can't be used when the economy is out of equilibrium (in a recession). That's non-ideal information transfer and I get to that in the last two paragraphs and the footnote below.

[2] This is non-ideal information transfer, and can be seen in an ideal gas where molecular forces become important (and the gas condenses into a liquid ... see link).

[3] I have a conjecture that this always happens.

Deaton's paradox and the dimension of consumption space

Today's economics Nobel brought up the "Deaton paradox" [pdf], that consumption is much smoother than income:

One of the most striking features of aggregate consumption behaviour is that aggregate consumption is smooth relative to aggregate income. Shifts in aggregate income are associated with relatively small shifts in aggregate consumption, and variations in consumption about trend are smaller than variations in income about trend.

You can read the paper for an explanation (or this paper for a different explanation).

However, in the information equilibrium view it is possible this is explained by the fact that there are more consumption dimensions (time periods and different options) than there are income dimensions. People change jobs less often than they buy food, and there are more distinguishable products out there than there are distinguishable jobs.

See here for how consumption smoothing is an emergent property of a representative agent.

At least, that's my first take upon hearing about it.

Economics as and versus social science

... or, Macro bee different from micro.

Bees!

My blurb on how I think that the whole "economics is too complex to make neat mathematical models" argument really just tends to assume itself (i.e. the original meaning of question begging) was picked up over at Mike Norman Economics, where it was put in an interesting way. Sociologists et al say economists need to prove economics is not too complex to model and economists say sociologists et al need to prove it is too complex to model.

That is to say we have sociologists and economists making a play for the null hypothesis.

My personal view is that economists should get the null hypothesis in this case, but not for reasons that economists think they should.

The key point is that (probably) the only way the actions of millions of complex humans can be aggregated in a tractable way is for the law of large numbers to kick in. In that case, the bulk properties of the state space tell us more about macroeconomic properties than the properties of the agents. Also in that case, as long as the social science of the agents meets some pretty general constraints, there is a separation between the economics (study of the state space) and the social science (study of the agents). The mathematical laws of economics are emergent and (somewhat) independent of the social agent substrate.

I borrowed a diagram from this post about how utility (a potentially measurable behavioral property of agents [1]) and entropy (a property of the bulk state space) can reproduce some of the same economic relationships ... and added bees.

The bees in the graphic at the top of this post represent the agents, and the observed dances and other behaviors are the social science theory of the agents. The bulk properties of the hive (size of cells, size of hive) represent the economic theory. We can tell some things from the law of large numbers ... the size of a cell varies a bit, but the average size (for a given species) can be well known ... the number of bees gives us a good estimate of the size of the hive. These give us a good idea of honey output regardless of the individual dances telling us the direction to some flowers with tasty nectar. Large scale coordinations break down these relationships (e.g. colony collapse) [2].

In my link above, there are some other things that give us more information. For example, in a d-dimensional beehive with d >> 1, nearly all the bees are near the surface, not the interior ... as long as they are not coordinated to be in the interior (say, by the queen).

When this separation holds, then economics is more like physics. When it doesn't, economics is a social science.

If the details of the complexity of bee social structure strongly mattered, it would (likely) be impossible to figure out how much honey you could get from N bees. Now humans are more complicated than bees, but the same principle -- that the macro properties are mostly governed by the bulk properties of the available state space -- has to apply if macro is tractable. And if it's not tractable (a possibility), then it really should just be moral and historical arguments.

Footnotes:

[1] ... that turns out to not really be true of individual agents in many experiments.

[2] This is the case of non-ideal information transfer in the model.

Pegged interest rates = hyperinflation?

Another data point for pegged interest rates leading to the hyperinflation solution from Brad DeLong. I didn't know that Weimar Germany had pegged interest rates.

I hope Angus Deaton gets a chance to cancel out Tom Sargent

Tom Sargent's old graduation speech about what econ (purportedly) teaches us came up again recently in my Twitter feed from Cameron Murray with a reference to the Nobel prize in economics making it topical for today.

Noah Smith did a pretty good job of going through it awhile ago from a more mainstream viewpoint. However, I thought I'd give it a information equilibrium perspective take (as well as my own opinions).

1. Many things that are desirable are not feasible.

This is kind of vague. The first thing that comes to mind for me is warp drive, but that has nothing to do with lessons from economics.

2. Individuals and communities face trade-offs.

But you really shouldn't assume this going into an economic analysis. Maybe there is a win-win; maybe there is a win-win in a given framework.

3. Other people have more information about their abilities, their efforts, and their preferences than you do.

When it comes to preferences, that's not really true -- at least in the technical economics definition of well-defined preferences. No one has good information about preferences because they are not necessarily stable nor transitive. Preferences may become well-defined, but only in aggregate.

4. Everyone responds to incentives, including people you want to help. That is why social safety nets don’t always end up working as intended.

I've written about incentives -- people don't respond to them so much as wander into the state space opened up by e.g. offering tax breaks. And a social safety net may be the key to allowing more people to explore that state space resulting in economic growth.

5. There are trade offs between equality and efficiency.

If information entropy is the key to economic growth, then more equality (higher entropy distribution) means more efficiency.

6. In an equilibrium of a game or an economy, people are satisfied with their choices. That is why it is difficult for well meaning outsiders to change things for better or worse.

This requires stable, transitive preferences (utility as a real-valued function). This is not necessarily a property of agents, but rather an emergent property. In that case, it wouldn't be true of individuals.

7. In the future, you too will respond to incentives. That is why there are some promises that you’d like to make but can’t. No one will believe those promises because they know that later it will not be in your interest to deliver. The lesson here is this: before you make a promise, think about whether you will want to keep it if and when your circumstances change. This is how you earn a reputation.

I covered incentives above in #4.

8. Governments and voters respond to incentives too. That is why governments sometimes default on loans and other promises that they have made.

This one is funny if you think about Greece because it predicts the exact opposite of what has happened: Greece stays in Euro and doesn't default on loans ... keeping promises when a) there are incentives not to keep them and b) keeping the promise is long run unsustainable.

9. It is feasible for one generation to shift costs to subsequent ones. That is what national government debts and the U.S. social security system do (but not the social security system of Singapore).

This is not only feasible, but desireable. Maximizing the number of effective time periods maximizes output, asset value and utility. See also #10.

10. When a government spends, its citizens eventually pay, either today or tomorrow, either through explicit taxes or implicit ones like inflation.

Actually, neither debt nor anything else seems to generate inflation when the information transfer index k ~ 1 unless the spending levels are so large as to change the relative size of nominal output and the monetary base (minus reserves) by a sizable fraction ... in logarithm. That is to say, big enough to change k. 

Also, Japan has been getting away with this for awhile.

11. Most people want other people to pay for public goods and government transfers (especially transfers to themselves).

Most people think of the government's budget as a household budget. Most people think the budget deficit gets worse when the President of the opposite political affiliation is in office. Most people aren't data-driven and don't really know how this stuff works. 

This one is especially funny; why would an economics professor give credence to what people who haven't studied the subject think? Does he think whatever it is that he learned is useless? Or did he learn that after thinking about it you should defer to "most people"?

I think most people think quantum processes are actually deterministic in some way. Should physicists take heed?

12. Because market prices aggregate traders’ information, it is difficult to forecast stock prices and interest rates and exchange rates.

The market doesn't always work as an information aggregation mechanism. And exchange rates and interest rates may be volatile, but aren't necessarily unpredictable.