Implausible plausibility

Likely in response to Paul Krugman's post titled "Plausibility", JW Mason put out his own post titled "Plausibility". However, the data shown is from 1925 to 2005 -- and includes the Great Depression and the mobilization for WWII:

Without those periods, the Friedman/Sanders point is well outside the mean (it is a 4.0-sigma deviation along the x-axis and a 2.8-sigma deviation along the y-axis) -- which was exactly Krugman's point:

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Update

JW Mason has updated his post with regard to this objection -- and thinks a WWII level of mobilization is possible. He now adds a linear fit ... but that fit likely leaves out some of the data (marked in red, including it gives us the red line):

But the other thing Mason suggests is that we can select a point along this line, which is basically an assumption about the current output gap and deviation from the trend.

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Update, the second

Unfortunately I had to attend to some errands, and so didn't complete my thought above. Additionally, Bill in comments below hits on something relevant that I want to fold in:

However, since Sanders's proposals are radically different from US government policies in the post-WWII period, why should we expect their results to be the same?

The key point to understand is that the US economy is assumed to have a log-linear RGDP per capita that holds from 1925 until today. John Handley discusses some of the issues with that assumption at his blog here (he sent a link in comments below) -- for example demographics.

Let's suspend our disbelief for a second and take it on its face: there is a trend of 2.3% log-linear RGDP per capita growth. This trend exists from the roaring 20s, though the depression, the New Deal, WWII's top-down industrial policy, the Great Society, deregulation, Reaganomics, the 90s tech boom, George W. Bush, the housing crisis, the financial crisis, the ARRA and Obamacare.

So is the trend the result of policy?

I noted before that the recovery in unemployment has a remarkable regularity:

As I said there:

This regularity over several decades would imply that any mechanism that explains the rate likely has nothing to do with the internet, inequality, jobless recoveries, war, government spending, unemployment benefits, Keynesianism, monetarism, technology, ... etc. It is doubtful these different forces conspire in differing degrees to achieve approximately the same result every time.

It would be doubtful that government policy has much impact on a trend growth that purportedly holds for a hundred years (1925 - 2026). An anonymous commenter below asks:

If Sanders is planning a WWII-like mobilization, why wouldn't the data from the last one and the years preceding it be relevant?

Mason seems to agree with the anonymous commenter; they believe this kind of mobilization is plausible (such that it should be the null hypothesis -- you have to explain why it is implausible).

However, they can't have it both ways. Either 1) there is reversion to a 2.3% trend that holds over a hundred years, making the Friedman/Sanders claims plausible, or 2) policy is relevant. If it is just reversion to trend, then Trump's policies would get us there too. The plausibility claim is either banal or a massive WWII-sized mobilization.

If policy is relevant and we could implement WWII-sized mobilizations whenever we'd like, then what is keeping that 2.3% log-linear growth trend in force over a hundred years? What steps in to say: Your mismanagement of the Great Depression has gone too far from trend. Please start a world war now. What steps in to say: Your WWII expansion gone too far above trend. Please end the war now. Did WWII start because the global depression was mismanaged? I have to say: that is an interesting theory. Maybe there is something to it. But the flipside that WWII ended because we had re-established real output per capita is kind of silly: we got tired of fighting and decided to make money instead.

I mentioned the other day that the Friedman/Sanders growth claims smack of neoliberalism -- neoclassical economics in service of progressive goals. A constant growth rate of 2.3% is basically a neoclassical growth model. In that view, we let the market do its dirty business, and growth happens. At least in that case it is explicit: it is a balancing of current consumption versus future consumption that creates an equilibrium growth rate.

These are two possible implicit theories behind the prima facie "model free" acceptance of the plausibility of a 2.3% growth rate.

I have a hard time accepting simultaneously a hundred-year-long 2.3% growth trend and relevant policy. With the IT model, I actually go for the former! There are significant long-run trends in the IT model in which policy plays little-to-no role. It's fairly empirically successful, but also says post-financial crisis growth is on-trend:

This isn't constant 2.3% RGDP per capita growth, though.

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Update, the third

If we extrapolate from our current (post-financial crisis) trend, we'd be here:

This is another way of illustrating that the assumed output gap is huge -- Friedman/Sanders takes us from the red dot to the orange dot.

You may ask why the red dot is so far from the data -- that's really an indication of how implausible the 2.3% growth trend is. If we assume a trend of 1.3% (red dot appears close to the data) or 3.3%, we get completely different figures:

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Update, the fourth (22 Feb 2016)

JW Mason stops by in comments below; I thought I'd illustrate the idea that the positive demand shocks have to balance the negative demand shocks to maintain a constant growth rate. I used an ARIMA process to simulate output. In one case I added large negative demand shocks (e.g. Great Depressions ... which are persistent in this model since trend growth will only slowly undo a large demand shock), and in another I added large positive demand shocks (e.g. WWII-mobilizations, which are also persistent).

Here is the first case:

And here is the second:

In the first case, the growth rate is on average lower with the large negative shocks (yellow) than without the shock (blue). In the second case,  the trend is on average about the same with the large balanced shocks (yellow) and without (blue).

The negative shocks correspond to the points labelled "Great Depression" and the positive shocks correspond to the points labelled "WWII" in dark blue above.

IT model forecast update for Japan

Here is an update of the data versus the forecast models for Japan. I am uncertain about how the BoJ arrived at their numbers for the annual inflation rates, so I opted for a agnostic approach that estimated the annual inflation rate using a linear model and took standard errors (2-sigma). Here is the result for core CPI inflation as was shown here and also a longer term version:

And here is my traditional format (for the CPI level forecast):

Extreme nominal wage rigidity

In one of Paul Krugman's presentations about macro [pdf], he presents a picture of nominal wage rigidity:

Is that 2012 figure for real? It's inconsistent with e.g. this result:

Also shown here [pdf]:

I would agree that is some serious nominal wage rigidity! However e.g. the US doesn't exhibit that magnitude of an effect, so it hardly seems like this would be a general conclusion.

The forecast's substrate: is Sanders condoning neoliberalism?

Central banks versus the IT model on inflation (link).

Steve Randy Waldman sends us to David Dayen and the push back against the push back against some optimistic projections (from Gerald Friedman) cited by the Sanders campaign (that I already made fun of here). Dayen asserts that economic forecasts are basically all garbage (and I agree), and so it is a bit disingenuous to selectively call out any particular one.

And I would agree -- if we were strictly talking about forecasting error. And if we were talking about forecasting error, everyone should kneel before Zod check out the IT model. However I am more interested in the implied model behind Friedman's forecast of Sanders policy.

The problem is that the substrate smacks of "neoliberalism". Making human progress and well-being all about economics is probably the best operational definition of "neoliberalism" (see here). The thinking seems to be that Bernie Sanders's policies are good therefore they should lead to good results (i.e. economic growth) ... and since they are so much better than other policies, they should lead to massive economic growth.

But growth is not always good; it depends on where it comes from.

No unit root?

JW Mason writes a blog post in defense of the estimate, building an argument that it is not entirely unreasonable. However, the first two points Mason makes are tied to a particular view of economic growth:

1. It’s not controversial to say that a historically deep recession ought to be followed by a period of historically strong growth.

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2. Friedman’s growth estimates are just what you need to get output and employment back to trend.

Actually, it is controversial to say that growth will return to trend -- it represents a claim that RGDP growth fluctuations do not have a unit root relative to the trend. In that view, economic shocks do not do long term damage to the economy ... and therefore there aren't hysteresis effects on employment. In a sense, this is a neoclassical growth model. Those models are completely mainstream (taught in economics classes), but not necessarily consistent with empirical data.

The IT model says RGDP growth fluctuations don't have a unit root, but that's relative to the IT model trend ... which calls for decreasing growth.

No housing bubble?

If the Sanders projections are just a return to trend, then that trend includes the housing bubble (Matthew Klein via Mason). A big part of the Sanders campaign is that millions of people lost their homes because of financial speculation ... and we bailed out the speculators. In general, the idea that bubbles don't exist is based on rational expectations. If housing was rationally priced in the 2000s, then there wasn't speculation.

The IT model says there was a housing bubble and it represented a deviation from the trend.

What is the source of the growth?

These projections don't seem to flow from the policies Sanders advocates. For example, much of the reduced labor force participation comes from an aging population. We don't want to make retirees re-enter the labor force. Matthew Yglesias also makes a great point: a lot of Sanders's policies should actually reduce labor force participation -- free college, better social security -- and this is a good thing! In the quantity theory of labor model, it would mean less economic growth ... which is not the be-all and end-all of human existence anyway.

In that sense, these growth projections seem more neoliberal than progressive. That's probably why they seem as rosy as Republican plans!

However, in the IT model, changes in the labor force are key to economic growth (and may be the only thing that ever has created economic growth). If Sanders can achieve the kind of increase in the labor force, then 5% RGDP growth isn't extraordinary -- we should really expect 10%. And it could be achieved with relaxed immigration policy, for example.

Are you saying the projections are attainable or not?

Neither. I am saying the projections aren't necessarily consistent with a progressive agenda. The projections seem to involve lots of working college students and senior citizens (or massive immigration). The 11 million undocumented immigrants Sanders wants to provide a path to citizenship for (with which I agree) are already here and generally working. We'd actually need 11 million additional immigrants to generate the scale of labor force increases above.

I see our future looking more like Japan -- low labor force growth (or even shrinkage) and likely deflation. These are bad for neoliberal growth-obsessed economics. And unless the US becomes amenable to vastly more immigration, what we need are policies that don't take economic growth as a given.

Right now the IT model projects linear growth (not log-linear ... linear-linear, see e.g. here), which means slowly decreasing growth rates:

This kind of tide raises no boats, and our institutions are completely unprepared to deal with it.

10% growthiness

I figured since I made fun of John Cochrane's defense of Jeb Bush's claim of 4% growth, I should maintain some balance and make fun of Bernie Sanders' economic proposals (see e.g. here or here). From the second link, we can get derive an estimate of the projected civilian labor force (CLF) and use the "quantity theory of labor" model to estimate the NGDP growth we'd expect. While the Sanders version only says NGDP growth would be 8.7% 5.3%, the labor force increases would actually imply an even higher NGDP growth of 10.5% (averaged over 2016-2026):

The dotted lines are the log-linear extrapolations for the IT model (blue) and the data (yellow), the projection based on the CLF figures derived from the Sanders campaign (green, BS) as well as the NGDP number of 5.3% from the campaign (red, BS₀). Note that this model also implies inflation would be 7.9% (and therefore average real growth would be 2.7%).

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Update 19 Feb 2016

I read the 5.3% as a nominal growth figure (you can see why I'd make such a mistake!); nominal growth in Gerald Friedman's projection to 2026 is more like 8.7% (to 43 trillion from 18 trillion) -- and therefore 3.4% inflation. Disregard the red line in the above graph. Edited the above post to reflect this.

Update 20 Feb 2016

Here is some good commentary at EconoSpeak on the projections themselves.

Update 22 Feb 2016

Corrected the red line (finally).

The value of diversity and upward sloping supply curves

Let me expand on something I said in this post:

There's another possible explanation of the bowed-out [Production Possibilities Frontier] curve. In the information equilibrium model, information entropy is equivalent to aggregate demand. Therefore the states with higher information entropy (i.e. states with more equal probability of finding apples and bananas) have higher AD relative to states with lower entropy (i.e. states with higher probability of finding either apples or bananas). Therefore AD near the middle of the PPF is slightly higher. This leads to a bowed-out PPF and upward sloping supply curves.

In these two posts ([1] and [2]), I show how to account for a contribution to output due to entropy (in terms of economic potentials, analogous to thermodynamic potentials). That is to say nominal output = sum of goods and services + entropy of goods and services. We don't know what the coefficient of the second term is exactly so I let it vary in the simulation below. We take the quantity of goods and services to be limited by a budget constraint (i.e. more X → X + dX means less Y → Y – dX), but allow that budget constraint to have a contribution due to entropy. There is a "real" budget constraint -- one more Xylophone means one less Yak -- but the nominal value of 5 Xylophones and 5 Yaks is greater than the nominal value of 10 Yaks or 10 Xylophones. By how much? I let that vary from zero to "a lot" in the simulation. One other thing to note is that I discussed this idea here in the context of Diane Coyle's review of Cesar Hidalgo's book Why Information Grows.

So here is the simulation. I generated 10,000 allocations of up to 50  yaks and xylophones (X + Y = 50) and added a constant (ranging from zero to "a lot") times the entropy of the resulting allocation to the total value of the yaks and xylophones ... and then normalized everything because the specific numbers don't matter. Here's the result (blue dots are the 10,000 allocations, the dashed straight line is the "real" budget constraint X + Y = 1 ... i.e. the prices of X and Y are equal, and the dashed curved line is the "real" budget constraint plus the entropy = PPF):

You can see that the entropy term creates a bowed-out PPF -- and thus upward sloping supply curves. The entropy term measures the value of diversity ... as Diane Coyle put it: a knife, a fork and a spoon is worth more than three spoons.

Recoveries do grow old; they just have an uncertain lifetime

There was an article out of the SF Fed that seems to suggest that post-WWII recoveries do not "grow old". They assume a Weibull distribution for the PDF for post-WWII recovery lengths (and I'm assuming for the pre-WWII recoveries as well) and then observe that the hazard function is "nearly flat" (has no time scale). The implication is that post-WWII recoveries don't "grow old" and end in recession.

I wasn't entirely convinced by that argument -- the reason for a hazard function to not be flat is that it has not only a well defined mean lifetime (scale), but additionally a very low variance. Those properties result in a strongly peaked PDF which leads to a sharply increasing hazard function over a short period of time. Because of the graph limits shown in the SF Fed article, it makes it look like the pre-WWII hazard function shoots up forever, while the post-WWII hazard function gradually increases. The pre-WWII hazard function actually has to level off after that initial increase -- which isn't shown.

The real argument that post-WWII recoveries don't grow old is that they are scale-free -- lack a well defined lifetime. However the use of a Weibull distribution for the post-WWII recoveries assumes they do have a lifetime (i.e. Weibull distribution has time scale).

But you can't assume a different type of distribution for the post-WWII recoveries than for the pre-WWII recoveries because you don't have very much post-WWII data. In a sense, you're kind of stuck. The data is mostly pre-WWII and a Weibull distribution looks appropriate for that. That means you're stuck with a Weibull distribution with different parameters for the post-WWII data [1] -- and therefore stuck with a lifetime.

I basically re-did the analysis using a gamma distribution to find out the scales (mean and variance). I fitted the empirical CDFs to regularized gamma functions and extracted the distributions for pre-WWII, post-WWII and used a sum of the distributions to represent all the data. Here are the resulting fits:

And here are the resulting PDFs:

The two distributions have means of 2.2 years (pre-WWII) and 4.8 years (post-WWII), but the big difference is in the variance. The square root of the variances are 1.0 years and 3.1 years. The post-WWII distributions are quite spread out. The result is a "flat" hazard function:

But post-WWII recoveries still "grow old" (after about 4.8 years), they just don't have a well-defined lifetime (± 3.1 years). As the current recovery is about 30 quarters long, so we'd expect about a 9% chance of a recession in the next quarter. But since the variance is so high, it takes 10 years (40 quarters) to reach a 99% chance of a recession (at least with this model):

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Footnotes

[1] Statistical tests pretty definitively show that the post-WWII data is drawn from a different distribution than the pre-WWII data. We know that the distribution changed, but we don't really have enough data to pick a new kind of distribution -- even though that might be warranted.

All that has plasmon frequencies in the visible is not 197Au

Image from wikimedia commons.

I recently downloaded a trial copy of Mankiw's Principles of Economics from Amazon looking for things to try and explain within the information equilibrium framework. After looking through it again to verify something for a comment at Noahpinion (that it doesn't have any real empirical results in it), I came across an entertaining "just so" story.

In the book is a story on why gold became a standard for money. They ask Sanat Kumar, a chemical engineer at Columbia, which element should be used for money. After throwing out all the reactive and radioactive metals, the gasses, the liquids, the metals with too high a melting point for ancient societies (platinum), and finally the metals that are too rare (osmium), he comes down to gold. It's stable; isn't reactive; isn't radioactive; it's rare, but not too rare.

The thing is that the unmentioned "on the periodic table" and the vague "rare, but not too rare" are really doing most of the work. Many metals react with the oxygen in the atmosphere to form an oxide — but it is usually only a thin surface (zinc oxide, aluminum oxide, copper oxide ... just check the change in your pocket). There aren't too many violently reacting metals (lithium through cesium). And various alloys could easily fit the bill.

This also leaves off several other desirable properties of money as well: not usable for other purposes (leading to consumption of the medium of exchange/store of value), not too heavy, not in finite supply (resulting in continuous deflation), and easily measurable (assaying metal or detecting counterfeit with fiat currency). 

Actually, the quantum properties of gold turn out to be important — the plasmon frequency of gold is in the visible spectrum making it yellowish as opposed to silvery like most metals. That property coupled with a touchstone gave ancient cultures a way of assaying the purity. Without that, no one could really trust the purity and therefore value of your gold — fulfilling the last property above.

In addition, most societies did not use gold as money — they used electrum, copper, bronze, cowrie shells, bronze cowrie shells, clay tokens, paper, etc. In some cases gold was only used for large transactions. In medieval Europe the rarity of gold (and silver and copper) likely inhibited economic development for hundreds of years (see also here) — yet rarity is listed as a desirable property in Kumar's list.

In reality, it was the combination of an incorrect theory of value (rarity) and an incorrect theory of macroeconomics (bullionism/mercantilism) that lead to gold becoming a standard.

PS There is also my pet theory: no intrinsic value and maximum information entropy. 

As if: positive economics, evolution and effective theories

David Sloan Wilson's latest piece at Evonomics brings up a good frame for my trilogy on production possibilities from this past week:

1. Production possibilities and the slope of the supply curve
2. Production possibilities and Brownian motion
3. Fitness, trade-offs and macrofoundations

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Positive economics

The argument I make at the end of [3] connects to this post about effective degrees of freedom near equilibria [4]: that rational agents are an effective theory for small perturbations around a macroeconomic equilibrium. I realize that Milton Friedman's The Methodology of Positive Economics [pdf for 1966 version, here for 1953 version] is an attempt to say that rational agents are an effective theory:

The abstract methodological issues we have been discussing have a direct bearing on the perennial criticism of "orthodox" economic theory as "unrealistic" as well as on the attempts that have been made to reformulate theory to meet this charge. Economics is a "dismal" science because it assumes man to be selfish and money-grubbing, "a lightning calculator of pleasures and pains, who oscillates like a homogeneous globule of desire of happiness under the impulse of stimuli that shift him about the area, but leave him intact" ... 

It is frequently convenient to present such a hypothesis by stating that the phenomena it is desired to predict behave in the world of observation as if they occurred in a hypothetical and highly simplified world containing only the forces that the hypothesis asserts to be important.

Emphasis in the original. The humans function as if they were rational agents is analogous to saying the theory of quarks and gluons acts as if it was a theory of color-neutral bosons called pions.

There are two problems with this. The first is the "asserts" in the last quoted sentence. You need some empirical fact from which to draw your effective theory. These empirical facts can lead to limits (Newtonian gravity is a limit of general relativity for slow speeds and low field strengths) or symmetries (the approximate chiral symmetry of QCD leads to the pion description). For example in economics, I've taken the limit of low inflation to show that the AD-AS model has an effective description using the IS-LM model and taken an approximate long run neutrality as a symmetry principle to motivate information equilibrium.

The second problem is that there are always boundaries on your effective theory (defined by a scale), and the effective degrees of freedom represent perturbations from some equilibrium. The pion description of QCD is effective between roughly the masses of quarks (m ~ a few MeV) and the QCD scale (chiral condensate or RG scale Λ ~ hundreds of MeV). So you have energies E where m << E << Λ. For E ~ GeV, you need to resort to QCD. For E much less than m, you can really just use quantum mechanics.

In [4] above, I argue that rational agents are an effective description only near a macroeconomic equilibrium. The limit in the information equilibrium description would be e.g. I(AD) ~ I(AS) -- i.e. approximate information equilibrium between aggregate demand aggregate supply. Therefore rational agents would only really apply outside of recessions.

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Positive biology

Sloan summarizes Friedman's analogies (n.b. these is in the 1953 version, not the 1966 version Sloan links to, which confused me at first):

Yet, [Friedman] claims that they are still predictive of human economic behavior by way of three analogies. First, trees distribute their leaves as if they are maximizing their exposure to sunlight, yet no one pretends that they are performing optimization equations. Likewise, an expert pool player acts as if he is performing complex calculations when making his shots, when in fact his behavior has been molded by countless hours of play. Finally, a firm acts as if it is maximizing its profits, when in fact its continuing survival is the result of a selection process in which the non-optimizing firms were eliminated.

Sloan further says these are evolutionary arguments [f1]:

The first is an example of genetic evolution, the second is an example of individual learning, and the third is an example of cultural evolution. In all cases, a process of selection results in entities that behave adaptively, as if they are solving complex optimization equations, when mechanistically they are doing nothing of the sort. ... 

So far, Friedman is standing on firm evolutionary ground with his “as if” argument [that agents are maximizing]. Evolutionists frequently reason about the properties of species “as if” they are maximizing their fitness, without worrying about the proximate mechanisms.

In a sense, maximizing fitness is an effective (an as if) theory of evolution. One way to achieve it (as I talk about in [3]) is to recognize that the maximizing state of that tree is actually just more likely than a non-maximizing state assuming there are a lot of traits that go into the exposure to sunlight (leaf size, shape, branching ratios, height, thickness of trunk and branches, etc ... ).

Sloan then provides some caveats via Gould [f2] and Lewontin; in particular the "as if" effective theory is only valid if you correctly identify the trait as the result of selection and identify the selection pressures. I don't agree with the latter piece (selection pressures aren't always necessary per [3] unless it is something e.g. controlled by a single gene), but the former is essentially the statement that the "as if" effective theory of maximized fitness is only valid in the neighborhood of an equilibrium (e.g. the sandy brown coloring of desert creatures). And that equilibrium requires "macrofoundations" [4] -- i.e. a stable ecosystem (the desert has been a desert for awhile).

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Summary

From this discussion we can conclude that assuming maximizing rational agents in economics during a recession is like assuming maximized fitness for species subject to climate change. If the changes (or recessions) are small, you might be able to assume maximization, but in general this is not a valid assumption.

We can construct a list of requirements for simple effective ("as if") descriptions in complex systems:

1. Macro-scale (i.e. system) equilibrium
2. Empirical facts on which to base your effective degrees of freedom
3. Scope conditions defining the neighborhood of validity for those effective degrees of freedom

Additionally, maximum entropy gives us some simplifications in cases where the number of dimensions is large.

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Footnotes:

Since I used [n] to indicate the blog post references above, the footnotes have the form [fn].

[f1] A minor quibble at this point: I don't think Friedman is actually talking about how trees or billiard players achieved their maximizing state, so he is not making an evolutionary argument. He is talking about describing a maximizing state in terms of detailed rules and calculations required to determine the maximizing state versus just saying it is a maximizing state. The billiard players behave as if they are doing complex physics calculations, and you can describe that effectively with either the physics calculations or assuming maximization -- even though neither is actually happening. Trees behave as if they are doing a complex optimization, and you can describe that effectively with a complex optimization or assuming maximization -- even though neither are happening. Sloan's interpretation is that you can describe these states either as maximization or as the result of an evolutionary process -- even though only the latter is happening. But Sloan's interpretation is useful for my purposes here, hence I put this in a footnote.

[f2] Here's Paul Krugman mentions Gould in the course of this article.

Fitness, trade-offs and macrofoundations

Nick Rowe sent me a link to one of his posts alongside one that used it for an analogy in evolutionary biology. They're both interesting reads, so check them out. I'll wait.

The basic idea is that competition should create a state for an economy or an organism where "fitness" should be maximized (the production possibility frontier saturated). In this state, any further change in production (or adaptations) should lead to trade offs. The organism or economy is at the boundary (set by e.g. energy or resource constraints), so the only states available to move into involve trade-offs.

Eagles as a species have really good eyesight, but can't hunt at night and economies produce the optimal amount of apples and bananas. Night hunting would come at the cost of day hunting and more apples comes at a cost of fewer bananas.

You can visualize both equilibria as the result of optimizing agents. Each eagle in the species tries to maximize food calories and offspring. Each firm in the economy tries to make the most money producing apples and/or bananas.

But you can also visualize both equilibria as the result of a random exploration of the state space -- especially when the fitness or production being optimized has a lot of dimensions (number of traits or number of products). Let me draw a picture:

As the boundary (production possibilities or available food energy) expands, the number of states in a swath near the boundary increases as the distance from the origin: n ~ Rᵈ⁻¹. Therefore the number of states at r₂ is greater than at r₁; if we take d >> 1 then most of the states will be near the boundary (in purple below).

And because of that, the economy or species will most likely be in a state near the boundary (e.g. the one highlighted in yellow) after a random walk (purple dots). Those purple dots (economy or species states) are like the pollen grains in Brownian motion with the actions of individual companies or inherited traits of individual organisms being the water molecules randomly moving the pollen from state to state (position to position). That boundary could be due to energy constraints or budget constraints, but it could also be due to insufficient diffusion time (r ~ √ t independent of d) to reach states that are further out.

If the reason for the boundary is a constraint (not time), then movements from state to state on the boundary involve trade-offs, while movements in the interior don't -- as illustrated in the next figure.

So we'd expect to see economies and species in states of maximized production and fitness, respectively, experiencing trade-offs between production possibilities and traits, respectively.

Update 23 April 2017: I want to add that most of the states are in that swath near the boundary, so even random movements are more likely to take you to another state in the swath (i.e. a trade-off) rather than inward (fewer states) or outward (probability of occupying a state falls as you move towards or past the boundary) very far. End update.

However, if the boundary is just time, then there really aren't trade offs. At least at the beginning of life, evolution (state space exploration) was probably limited by the mutation rate. Sexual reproduction (or gene transfer) seems to have enhanced the rate of spreading helpful adaptations, speeding up the state space exploration. But early organisms might not have experienced trade-offs because the state space was wide open. Sure, deleterious mutations would die out, but advantageous mutations did not necessarily come at the expense of another adaptation. However after billions of years, we should probably expect trade-offs between many adaptations.

What about trade-offs in economics? Nick Rowe points out that sometimes there aren't trade-offs (exemplified with fuel injection), but says most of the time we should expect them in a sense for similar reasons in biology: competition and grabbing free lunches. However I'd like to connect this discussion to David Glasner's macrofoundations.

What determines the total amount of apples and bananas that can be produced in an economy? The number of people willing to buy and the money available for them to purchase them at the equilibrium prices. By another name: aggregate demand. The PPF for all goods and services is bounded by the total aggregate demand. In the long run, AD grows with time -- meaning that the boundary is just time. In this case there are no trade-offs. The economy can start to just produce more bananas with its economic growth without reducing the number of apples.

In the short run, however, there is an equilibrium level of AD (e.g. the crossing point of an AD-AS model diagram). In this case, there are trade-offs. There is a trade-off even for a linear boundary (e.g. a budget constraint) but let's consider the bowed-out curves illustrated in the pictures above -- corresponding to upward sloping supply curves. Rowe explains the bowed-put PPF in terms of comparative advantage or differences in utilization of factors of production (e.g. apples are more labor intensive than bananas).

There's another possible explanation of the bowed-out curve. In the information equilibrium model, information entropy is equivalent to aggregate demand. Therefore the states with higher information entropy (i.e. states with more equal probability of finding apples and bananas) have higher AD relative to states with lower entropy (i.e. states with higher probability of finding either apples or bananas). Therefore AD near the middle of the PPF is slightly higher. This leads to a bowed-out PPF and upward sloping supply curves.

Regardless of how you obtain it, the bowed-out curve -- plus the trade-offs and maximization at the (short run) equilibrium value of AD -- can be effectively described in terms of rational maximizing agents. But all of those properties follow from the macroeconomic conditions [and large number of products in the macroeconomy] describing the existence of a boundary [and giving reasons for being near it]. Without that bowed-out PPF, there are no trade-offs, no upward sloping supply curves, and no maximizing. These properties of rational agents are dependent on macrofoundations! And they only exist for small perturbations around the PPF established by that short run equilibrium value of aggregate demand.

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PS I'm not claiming originality here (much is borrowed from Nick Rowe), except for the d >> 1 explanation for maximization and the information entropy explanation of the bowed-out AD curve. I was unable to find any references for this particular take on macrofoundations of rational maximizing behavior, but that doesn't mean they don't exist.

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Update 13 Feb 2016

Added a couple sentences to the paragraph on evolution (and the one after it), which seemed a bit of a non sequitur without them. I wrote most of this on my flight home last night, and I forgot to finish up that bit between getting off the plane and posting it on the blog.