nhaliday + limits   67

multivariate analysis - Is it possible to have a pair of Gaussian random variables for which the joint distribution is not Gaussian? - Cross Validated
The bivariate normal distribution is the exception, not the rule!

It is important to recognize that "almost all" joint distributions with normal marginals are not the bivariate normal distribution. That is, the common viewpoint that joint distributions with normal marginals that are not the bivariate normal are somehow "pathological", is a bit misguided.

Certainly, the multivariate normal is extremely important due to its stability under linear transformations, and so receives the bulk of attention in applications.

note: there is a multivariate central limit theorem, so those such applications have no problem
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october 2017 by nhaliday
Karl Pearson and the Chi-squared Test
Pearson's paper of 1900 introduced what subsequently became known as the chi-squared test of goodness of fit. The terminology and allusions of 80 years ago create a barrier for the modern reader, who finds that the interpretation of Pearson's test procedure and the assessment of what he achieved are less than straightforward, notwithstanding the technical advances made since then. An attempt is made here to surmount these difficulties by exploring Pearson's relevant activities during the first decade of his statistical career, and by describing the work by his contemporaries and predecessors which seem to have influenced his approach to the problem. Not all the questions are answered, and others remain for further study.

original paper: http://www.economics.soton.ac.uk/staff/aldrich/1900.pdf

How did Karl Pearson come up with the chi-squared statistic?: https://stats.stackexchange.com/questions/97604/how-did-karl-pearson-come-up-with-the-chi-squared-statistic
He proceeds by working with the multivariate normal, and the chi-square arises as a sum of squared standardized normal variates.

You can see from the discussion on p160-161 he's clearly discussing applying the test to multinomial distributed data (I don't think he uses that term anywhere). He apparently understands the approximate multivariate normality of the multinomial (certainly he knows the margins are approximately normal - that's a very old result - and knows the means, variances and covariances, since they're stated in the paper); my guess is that most of that stuff is already old hat by 1900. (Note that the chi-squared distribution itself dates back to work by Helmert in the mid-1870s.)

Then by the bottom of p163 he derives a chi-square statistic as "a measure of goodness of fit" (the statistic itself appears in the exponent of the multivariate normal approximation).

He then goes on to discuss how to evaluate the p-value*, and then he correctly gives the upper tail area of a χ212χ122 beyond 43.87 as 0.000016. [You should keep in mind, however, that he didn't correctly understand how to adjust degrees of freedom for parameter estimation at that stage, so some of the examples in his papers use too high a d.f.]
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october 2017 by nhaliday
Section 10 Chi-squared goodness-of-fit test.
- pf that chi-squared statistic for Pearson's test (multinomial goodness-of-fit) actually has chi-squared distribution asymptotically
- the gotcha: terms Z_j in sum aren't independent
- solution:
- compute the covariance matrix of the terms to be E[Z_iZ_j] = -sqrt(p_ip_j)
- note that an equivalent way of sampling the Z_j is to take a random standard Gaussian and project onto the plane orthogonal to (sqrt(p_1), sqrt(p_2), ..., sqrt(p_r))
- that is equivalent to just sampling a Gaussian w/ 1 less dimension (hence df=r-1)
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october 2017 by nhaliday
Any particular gene has a specific location (its "locus") on a particular chromosome. For any two genes (or loci) alpha and beta, we can ask "What is the recombination frequency between them?" If the genes are on different chromosomes, the answer is 50% (independent assortment). If the two genes are on the same chromosome, the recombination frequency will be somewhere in the range from 0 to 50%. The "map unit" (1 cM) is the genetic map distance that corresponds to a recombination frequency of 1%. In large chromosomes, the cumulative map distance may be much greater than 50cM, but the maximum recombination frequency is 50%. Why? In large chromosomes, there is enough length to allow for multiple cross-overs, so we have to ask what result we expect for random multiple cross-overs.

1. How is it that random multiple cross-overs give the same result as independent assortment?

Figure 5.12 shows how the various double cross-over possibilities add up, resulting in gamete genotype percentages that are indistinguisable from independent assortment (50% parental type, 50% non-parental type). This is a very important figure. It provides the explanation for why genes that are far apart on a very large chromosome sort out in crosses just as if they were on separate chromosomes.

2. Is there a way to measure how close together two crossovers can occur involving the same two chromatids? That is, how could we measure whether there is spacial "interference"?

Figure 5.13 shows how a measurement of the gamete frequencies resulting from a "three point cross" can answer this question. If we would get a "lower than expected" occurrence of recombinant genotypes aCb and AcB, it would suggest that there is some hindrance to the two cross-overs occurring this close together. Crosses of this type in Drosophila have shown that, in this organism, double cross-overs do not occur at distances of less than about 10 cM between the two cross-over sites. ( Textbook, page 196. )

3. How does all of this lead to the "mapping function", the mathematical (graphical) relation between the observed recombination frequency (percent non-parental gametes) and the cumulative genetic distance in map units?

Figure 5.14 shows the result for the two extremes of "complete interference" and "no interference". The situation for real chromosomes in real organisms is somewhere between these extremes, such as the curve labelled "interference decreasing with distance".
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october 2017 by nhaliday
Lecture 14: When's that meteor arriving
- Meteors as a random process
- Limiting approximations
- Derivation of the Exponential distribution
- Derivation of the Poisson distribution
- A "Poisson process"
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september 2017 by nhaliday
Lucio Russo - Wikipedia
In The Forgotten Revolution: How Science Was Born in 300 BC and Why It Had to Be Reborn (Italian: La rivoluzione dimenticata), Russo promotes the belief that Hellenistic science in the period 320-144 BC reached heights not achieved by Classical age science, and proposes that it went further than ordinarily thought, in multiple fields not normally associated with ancient science.

La Rivoluzione Dimenticata (The Forgotten Revolution), Reviewed by Sandro Graffi: http://www.ams.org/notices/199805/review-graffi.pdf

Before turning to the question of the decline of Hellenistic science, I come back to the new light shed by the book on Euclid’s Elements and on pre-Ptolemaic astronomy. Euclid’s definitions of the elementary geometric entities—point, straight line, plane—at the beginning of the Elements have long presented a problem.7 Their nature is in sharp contrast with the approach taken in the rest of the book, and continued by mathematicians ever since, of refraining from defining the fundamental entities explicitly but limiting themselves to postulating the properties which they enjoy. Why should Euclid be so hopelessly obscure right at the beginning and so smooth just after? The answer is: the definitions are not Euclid’s. Toward the beginning of the second century A.D. Heron of Alexandria found it convenient to introduce definitions of the elementary objects (a sign of decadence!) in his commentary on Euclid’s Elements, which had been written at least 400 years before. All manuscripts of the Elements copied ever since included Heron’s definitions without mention, whence their attribution to Euclid himself. The philological evidence leading to this conclusion is quite convincing.8


What about the general and steady (on the average) impoverishment of Hellenistic science under the Roman empire? This is a major historical problem, strongly tied to the even bigger one of the decline and fall of the antique civilization itself. I would summarize the author’s argument by saying that it basically represents an application to science of a widely accepted general theory on decadence of antique civilization going back to Max Weber. Roman society, mainly based on slave labor, underwent an ultimately unrecoverable crisis as the traditional sources of that labor force, essentially wars, progressively dried up. To save basic farming, the remaining slaves were promoted to be serfs, and poor free peasants reduced to serfdom, but this made trade disappear. A society in which production is almost entirely based on serfdom and with no trade clearly has very little need of culture, including science and technology. As Max Weber pointed out, when trade vanished, so did the marble splendor of the ancient towns, as well as the spiritual assets that went with it: art, literature, science, and sophisticated commercial laws. The recovery of Hellenistic science then had to wait until the disappearance of serfdom at the end of the Middle Ages. To quote Max Weber: “Only then with renewed vigor did the old giant rise up again.”


The epilogue contains the (rather pessimistic) views of the author on the future of science, threatened by the apparent triumph of today’s vogue of irrationality even in leading institutions (e.g., an astrology professorship at the Sorbonne). He looks at today’s ever-increasing tendency to teach science more on a fideistic than on a deductive or experimental basis as the first sign of a decline which could be analogous to the post-Hellenistic one.

Praising Alexandrians to excess: https://sci-hub.tw/10.1088/2058-7058/17/4/35
The Economic Record review: https://sci-hub.tw/10.1111/j.1475-4932.2004.00203.x

listed here: https://pinboard.in/u:nhaliday/b:c5c09f2687c1

Was Roman Science in Decline? (Excerpt from My New Book): https://www.richardcarrier.info/archives/13477
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may 2017 by nhaliday
Mixing (mathematics) - Wikipedia
One way to describe this is that strong mixing implies that for any two possible states of the system (realizations of the random variable), when given a sufficient amount of time between the two states, the occurrence of the states is independent.

Mixing coefficient is
α(n) = sup{|P(A∪B) - P(A)P(B)| : A in σ(X_0, ..., X_{t-1}), B in σ(X_{t+n}, ...), t >= 0}
for σ(...) the sigma algebra generated by those r.v.s.

So it's a notion of total variational distance between the true distribution and the product distribution.
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february 2017 by nhaliday
Kolmogorov's zero–one law - Wikipedia
In probability theory, Kolmogorov's zero–one law, named in honor of Andrey Nikolaevich Kolmogorov, specifies that a certain type of event, called a tail event, will either almost surely happen or almost surely not happen; that is, the probability of such an event occurring is zero or one.

tail events include limsup E_i
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february 2017 by nhaliday
What is the relationship between information theory and Coding theory? - Quora
- finite vs. asymptotic
- combinatorial vs. probabilistic (lotsa overlap their)
- worst-case (Hamming) vs. distributional (Shannon)

Information and coding theory most often appear together in the subject of error correction over noisy channels. Historically, they were born at almost exactly the same time - both Richard Hamming and Claude Shannon were working at Bell Labs when this happened. Information theory tends to heavily use tools from probability theory (together with an "asymptotic" way of thinking about the world), while traditional "algebraic" coding theory tends to employ mathematics that are much more finite sequence length/combinatorial in nature, including linear algebra over Galois Fields. The emergence in the late 90s and first decade of 2000 of codes over graphs blurred this distinction though, as code classes such as low density parity check codes employ both asymptotic analysis and random code selection techniques which have counterparts in information theory.

They do not subsume each other. Information theory touches on many other aspects that coding theory does not, and vice-versa. Information theory also touches on compression (lossy & lossless), statistics (e.g. large deviations), modeling (e.g. Minimum Description Length). Coding theory pays a lot of attention to sphere packing and coverings for finite length sequences - information theory addresses these problems (channel & lossy source coding) only in an asymptotic/approximate sense.
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february 2017 by nhaliday
Information Processing: Epistasis vs additivity
On epistasis: why it is unimportant in polygenic directional selection: http://rstb.royalsocietypublishing.org/content/365/1544/1241.short
- James F. Crow

The Evolution of Multilocus Systems Under Weak Selection: http://www.genetics.org/content/genetics/134/2/627.full.pdf
- Thomas Nagylaki

Data and Theory Point to Mainly Additive Genetic Variance for Complex Traits: http://journals.plos.org/plosgenetics/article?id=10.1371/journal.pgen.1000008
The relative proportion of additive and non-additive variation for complex traits is important in evolutionary biology, medicine, and agriculture. We address a long-standing controversy and paradox about the contribution of non-additive genetic variation, namely that knowledge about biological pathways and gene networks imply that epistasis is important. Yet empirical data across a range of traits and species imply that most genetic variance is additive. We evaluate the evidence from empirical studies of genetic variance components and find that additive variance typically accounts for over half, and often close to 100%, of the total genetic variance. We present new theoretical results, based upon the distribution of allele frequencies under neutral and other population genetic models, that show why this is the case even if there are non-additive effects at the level of gene action. We conclude that interactions at the level of genes are not likely to generate much interaction at the level of variance.
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february 2017 by nhaliday
general topology - What should be the intuition when working with compactness? - Mathematics Stack Exchange

The situation with compactness is sort of like the above. It turns out that finiteness, which you think of as one concept (in the same way that you think of "Foo" as one concept above), is really two concepts: discreteness and compactness. You've never seen these concepts separated before, though. When people say that compactness is like finiteness, they mean that compactness captures part of what it means to be finite in the same way that shortness captures part of what it means to be Foo.


As many have said, compactness is sort of a topological generalization of finiteness. And this is true in a deep sense, because topology deals with open sets, and this means that we often "care about how something behaves on an open set", and for compact spaces this means that there are only finitely many possible behaviors.


Compactness does for continuous functions what finiteness does for functions in general.

If a set A is finite then every function f:A→R has a max and a min, and every function f:A→R^n is bounded. If A is compact, the every continuous function from A to R has a max and a min and every continuous function from A to R^n is bounded.

If A is finite then every sequence of members of A has a subsequence that is eventually constant, and "eventually constant" is the only kind of convergence you can talk about without talking about a topology on the set. If A is compact, then every sequence of members of A has a convergent subsequence.
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january 2017 by nhaliday
Mikhail Leonidovich Gromov - Wikipedia
Gromov's style of geometry often features a "coarse" or "soft" viewpoint, analyzing asymptotic or large-scale properties.

Gromov is also interested in mathematical biology,[11] the structure of the brain and the thinking process, and the way scientific ideas evolve.[8]
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january 2017 by nhaliday
Information Processing: Is science self-correcting?
A toy model of the dynamics of scientific research, with probability distributions for accuracy of experimental results, mechanisms for updating of beliefs by individual scientists, crowd behavior, bounded cognition, etc. can easily exhibit parameter regions where progress is limited (one could even find equilibria in which most beliefs held by individual scientists are false!). Obviously the complexity of the systems under study and the quality of human capital in a particular field are important determinants of the rate of progress and its character.
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january 2017 by nhaliday
Soft analysis, hard analysis, and the finite convergence principle | What's new
It is fairly well known that the results obtained by hard and soft analysis respectively can be connected to each other by various “correspondence principles” or “compactness principles”. It is however my belief that the relationship between the two types of analysis is in fact much closer[3] than just this; in many cases, qualitative analysis can be viewed as a convenient abstraction of quantitative analysis, in which the precise dependencies between various finite quantities has been efficiently concealed from view by use of infinitary notation. Conversely, quantitative analysis can often be viewed as a more precise and detailed refinement of qualitative analysis. Furthermore, a method from hard analysis often has some analogue in soft analysis and vice versa, though the language and notation of the analogue may look completely different from that of the original. I therefore feel that it is often profitable for a practitioner of one type of analysis to learn about the other, as they both offer their own strengths, weaknesses, and intuition, and knowledge of one gives more insight[4] into the workings of the other. I wish to illustrate this point here using a simple but not terribly well known result, which I shall call the “finite convergence principle” (thanks to Ben Green for suggesting this name; Jennifer Chayes has also suggested the “metastability principle”). It is the finitary analogue of an utterly trivial infinitary result – namely, that every bounded monotone sequence converges – but sometimes, a careful analysis of a trivial result can be surprisingly revealing, as I hope to demonstrate here.
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january 2017 by nhaliday
The infinitesimal model | bioRxiv
Our focus here is on the infinitesimal model. In this model, one or several quantitative traits are described as the sum of a genetic and a non-genetic component, the first being distributed as a normal random variable centred at the average of the parental genetic components, and with a variance independent of the parental traits. We first review the long history of the infinitesimal model in quantitative genetics. Then we provide a definition of the model at the phenotypic level in terms of individual trait values and relationships between individuals, but including different evolutionary processes: genetic drift, recombination, selection, mutation, population structure, ... We give a range of examples of its application to evolutionary questions related to stabilising selection, assortative mating, effective population size and response to selection, habitat preference and speciation. We provide a mathematical justification of the model as the limit as the number M of underlying loci tends to infinity of a model with Mendelian inheritance, mutation and environmental noise, when the genetic component of the trait is purely additive. We also show how the model generalises to include epistatic effects. In each case, by conditioning on the pedigree relating individuals in the population, we incorporate arbitrary selection and population structure. We suppose that we can observe the pedigree up to the present generation, together with all the ancestral traits, and we show, in particular, that the genetic components of the individual trait values in the current generation are indeed normally distributed with a variance independent of ancestral traits, up to an error of order M^{-1/2}. Simulations suggest that in particular cases the convergence may be as fast as 1/M.

published version:
The infinitesimal model: Definition, derivation, and implications: https://sci-hub.tw/10.1016/j.tpb.2017.06.001

Commentary: Fisher’s infinitesimal model: A story for the ages: http://www.sciencedirect.com/science/article/pii/S0040580917301508?via%3Dihub
This commentary distinguishes three nested approximations, referred to as “infinitesimal genetics,” “Gaussian descendants” and “Gaussian population,” each plausibly called “the infinitesimal model.” The first and most basic is Fisher’s “infinitesimal” approximation of the underlying genetics – namely, many loci, each making a small contribution to the total variance. As Barton et al. (2017) show, in the limit as the number of loci increases (with enough additivity), the distribution of genotypic values for descendants approaches a multivariate Gaussian, whose variance–covariance structure depends only on the relatedness, not the phenotypes, of the parents (or whether their population experiences selection or other processes such as mutation and migration). Barton et al. (2017) call this rigorously defensible “Gaussian descendants” approximation “the infinitesimal model.” However, it is widely assumed that Fisher’s genetic assumptions yield another Gaussian approximation, in which the distribution of breeding values in a population follows a Gaussian — even if the population is subject to non-Gaussian selection. This third “Gaussian population” approximation, is also described as the “infinitesimal model.” Unlike the “Gaussian descendants” approximation, this third approximation cannot be rigorously justified, except in a weak-selection limit, even for a purely additive model. Nevertheless, it underlies the two most widely used descriptions of selection-induced changes in trait means and genetic variances, the “breeder’s equation” and the “Bulmer effect.” Future generations may understand why the “infinitesimal model” provides such useful approximations in the face of epistasis, linkage, linkage disequilibrium and strong selection.
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january 2017 by nhaliday
ho.history overview - Proofs that require fundamentally new ways of thinking - MathOverflow
my favorite:
Although this has already been said elsewhere on MathOverflow, I think it's worth repeating that Gromov is someone who has arguably introduced more radical thoughts into mathematics than anyone else. Examples involving groups with polynomial growth and holomorphic curves have already been cited in other answers to this question. I have two other obvious ones but there are many more.

I don't remember where I first learned about convergence of Riemannian manifolds, but I had to laugh because there's no way I would have ever conceived of a notion. To be fair, all of the groundwork for this was laid out in Cheeger's thesis, but it was Gromov who reformulated everything as a convergence theorem and recognized its power.

Another time Gromov made me laugh was when I was reading what little I could understand of his book Partial Differential Relations. This book is probably full of radical ideas that I don't understand. The one I did was his approach to solving the linearized isometric embedding equation. His radical, absurd, but elementary idea was that if the system is sufficiently underdetermined, then the linear partial differential operator could be inverted by another linear partial differential operator. Both the statement and proof are for me the funniest in mathematics. Most of us view solving PDE's as something that requires hard work, involving analysis and estimates, and Gromov manages to do it using only elementary linear algebra. This then allows him to establish the existence of isometric embedding of Riemannian manifolds in a wide variety of settings.
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january 2017 by nhaliday
Breeding the breeder's equation - Gene Expression
- interesting fact about normal distribution: when thresholding Gaussian r.v. X ~ N(0, σ^2) at X > 0, the new mean μ_s satisfies μ_s = pdf(X,t)/(1-cdf(X,t)) σ^2
- follows from direct calculation (any deeper reason?)
- note (using Taylor/asymptotic expansion of complementary error function) that this is Θ(t) as t -> 0 or ∞ (w/ different constants)
- for X ~ N(0, 1), can calculate 0 = cdf(X, t)μ_<t + (1-cdf(X, t))μ_>t => μ_<t = -pdf(X, t)/cdf(X, t)
- this declines quickly w/ t (like e^{-t^2/2}). as t -> 0, it goes like -sqrt(2/pi) + higher-order terms ~ -0.8.

Average of a tail of a normal distribution: https://stats.stackexchange.com/questions/26805/average-of-a-tail-of-a-normal-distribution

Truncated normal distribution: https://en.wikipedia.org/wiki/Truncated_normal_distribution
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december 2016 by nhaliday
Borel–Cantelli lemma - Wikipedia
- sum of probabilities finite => a.s. only finitely many occur
- "<=" w/ some assumptions (pairwise independence)
- classic result from CS 150 (problem set 1)
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november 2016 by nhaliday
Fiction: Missile Gap by Charles Stross — Subterranean Press
- flat-earth scifi
- little tidbit from fictional Carl Sagan: behavior of gravity on very large (near-infinite) disk
in limit, no inverse square law, constant downward force: ∫ G/(a^2+r^2) a/sqrt(a^2+r^2) σ rdr dθ = 2πGσ independent of a
for large but finite radius R, asymptotically inverse square but near-constant for a << R (check via Maclaurin expansion around a and x=1/a)
- interesting depiction of war between eusocial species and humans (humans lose)
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october 2016 by nhaliday
Why Information Grows – Paul Romer
thinking like a physicist:

The key element in thinking like a physicist is being willing to push simultaneously to extreme levels of abstraction and specificity. This sounds paradoxical until you see it in action. Then it seems obvious. Abstraction means that you strip away inessential detail. Specificity means that you take very seriously the things that remain.

Abstraction vs. Radical Specificity: https://paulromer.net/abstraction-vs-radical-specificity/
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september 2016 by nhaliday
Too much of a good thing | The Economist
None of these accounts, though, explain the most troubling aspect of America’s profit problem: its persistence. Business theory holds that firms can at best enjoy only temporary periods of “competitive advantage” during which they can rake in cash. After that new companies, inspired by these rich pickings, will pile in to compete away those fat margins, bringing prices down and increasing both employment and investment. It’s the mechanism behind Adam Smith’s invisible hand.

In America that hand seems oddly idle. An American firm that was very profitable in 2003 (one with post-tax returns on capital of 15-25%, excluding goodwill) had an 83% chance of still being very profitable in 2013; the same was true for firms with returns of over 25%, according to McKinsey, a consulting firm. In the previous decade the odds were about 50%. The obvious conclusion is that the American economy is too cosy for incumbents.

Corporations Are Raking In Record Profits, But Workers Aren’t Seeing Much of It: http://www.motherjones.com/kevin-drum/2017/07/corporations-are-raking-in-record-profits-but-workers-arent-seeing-much-of-it/
Even Goldman Sachs thinks monopolies are pillaging American consumers: http://theweek.com/articles/633101/even-goldman-sachs-thinks-monopolies-are-pillaging-american-consumers
Schumpeter: The University of Chicago worries about a lack of competition: http://www.economist.com/news/business/21720657-its-economists-used-champion-big-firms-mood-has-shifted-university-chicago
Some radicals argue that the government is now so rotten that America is condemned to perpetual oligarchy and inequality. Political support for more competition is worryingly hard to find. Donald Trump has a cabinet of tycoons and likes to be chummy with bosses. The Republicans have become the party of incumbent firms, not of free markets or consumers. Too many Democrats, meanwhile, don’t trust markets and want the state to smother them in red tape, which hurts new entrants.

The Rise of Market Power and the Decline of Labor’s Share: https://promarket.org/rise-market-power-decline-labors-share/
A new paper by Jan De Loecker (of KU Leuven and Princeton University) and Jan Eeckhout (of the Barcelona Graduate School of Economics UPF and University College London) echoes these results, arguing that the decline of both the labor and capital shares, as well as the decline in low-skilled wages and other economic trends, have been aided by a significant increase in markups and market power.


Measuring markups, De Loecker explained in a conversation with ProMarket, is notoriously difficult due to the scarcity of data. In attempting to track markups across a wide set of firms and industries, De Loecker and Eeckhout diverged from the standard way in which Industrial Organization economists look at markups, the so-called “demand approach,” which requires a lot of data on consumer demand (prices, quantities, characteristics of products) and models of how firms compete. The standard approach, explains De Loecker, works when it is tailor-made for particular markets, but is “not feasible” when studying markups across many markets and over a long period of time.

To do that, De Loecker and Eeckhout use another approach, the “production approach,” which relies on standard, publicly-available balance sheet data and an assumption that firms will try to minimize costs, and does not require other assumptions regarding demand and market competition.


Markups, De Loecker and Eeckhout note, do not necessarily imply market power—but profits do. The enormous increase in profits over the past 35 years, they argue, is consistent with an increase in market power. “In perfect competition, your costs and total sales are identical, because there’s no difference between price and marginal costs. The extent to which these two numbers—the sales-to-wage bill and total-costs-to-wage bill—start differing is going to be immediately indicative of the market power,” says De Loecker.

Markup increases, De Loecker and Eeckhout find, became more pronounced following the 2000 and 2008 recessions. Curiously, they find that economy-wide it is mainly smaller firms that have the higher markups, which according to De Loecker is indicative of widely different characteristics between various industries. Within narrowly defined industries, however, the standard prediction holds: firms with larger market shares have higher markups as well. “Most of the action happens within industries, where we see the big guys getting bigger and their markups increase,” De Loecker explains.


The authors are correct that this can easily account for the apparent US productivity slowdown. Holding real productivity constant, if firms move up their demand curves to sell less at a higher prices, then total output, and measured GDP, get smaller. Their numerical estimates suggest that, correcting for this effect, there has been no decline in US productivity growth since 1965. That’s a pretty big deal.

Accepting the main result that markups have been marching upward, the obvious question to ask is: why? But first, let’s review some clues from the paper. First, while industries with smaller firms tend to have higher markups, within each small industry, bigger firms have larger markups, and firms with higher markups pay higher dividends.

There has been little change in output elasticity, i.e., the rate at which variable costs change with the quantity of units produced. (So this isn’t about new scale economies.) There has also been little change in the bottom half of the distribution of markups; the big change has been a big stretching in the upper half. Markups have increased more in larger industries, and the main change has been within industries, rather than a changing mix of industries in the economy. The fractions of income going to labor and to tangible capital have fallen, and firms respond less than they once did to wage changes. Firm accounting profits as a fraction of total income have risen four fold since 1980.


If, like me, you buy the standard “free entry” argument for zero expected economic profits of early entrants, then the only remaining possible explanation is an increase in fixed costs relative to variable costs. Now as the paper notes, the fall in tangible capital spending and the rise in accounting profits suggests that this isn’t so much about short-term tangible fixed costs, like the cost to buy machines. But that still leaves a lot of other possible fixed costs, including real estate, innovation, advertising, firm culture, brand loyalty and prestige, regulatory compliance, and context specific training. These all require long term investments, and most of them aren’t tracked well by standard accounting systems.

I can’t tell well which of these fixed costs have risen more, though hopefully folks will collect enough data on these to see which ones correlate strongest with the industries and firms where markups have most risen. But I will invoke a simple hypothesis that I’ve discussed many times, which predicts a general rise of fixed costs: increasing wealth leading to stronger tastes for product variety. Simple models of product differentiation say that as customers care more about getting products nearer to their ideal point, more products are created and fixed costs become a larger fraction of total costs.

Note that increasing product variety is consistent with increasing concentration in a smaller number of firms, if each firm offers many more products and services than before.



Variable costs approach zero: http://www.arnoldkling.com/blog/variable-costs-approach-zero/
4. My guess is that, if anything, the two-Jan’s paper understates the trend toward high markups. That is because my guess is that most corporate data allocates more labor to variable cost than really belongs there. Garett Jones pointed out that these days most workers do not produce widgets. Instead, they produce organizational capital. Garett Jones workers are part of overhead, not variable cost.

Intangible investment and monopoly profits: http://marginalrevolution.com/marginalrevolution/2017/09/intangible-investment-monopoly-profits.html
I’ve been reading the forthcoming Capitalism Without Capital: The Rise of the Intangible Economy, by Jonathan Haskel and Stian Westlake, which is one of this year’s most important and stimulating economic reads (I can’t say it is Freakonomics-style fun, but it is well-written relative to the nature of its subject matter.)

The book offers many valuable theoretical points and also observations about data. And note that intangible capital used to be below 30 percent of the S&P 500 in the 70s, now it is about 84 percent. That’s a big increase, and yet the topic just isn’t discussed that much (I cover it a bit in The Complacent Class, as a possible source of increase in business risk-aversion).


Now, I’ve put that all into my language and framing, rather than theirs. In any case, I suspect that many of the recent puzzles about mark-ups and monopoly power are in some way tied to the nature of intangible capital, and the rising value of intangible capital.

The one-sentence summary of my takeaway might be: Cross-business technology externalities help explain the mark-up, market power, and profitability puzzles.

Why has investment been weak?: http://marginalrevolution.com/marginalrevolution/2017/12/why-has-investment-been-weak.html
We analyze private fixed investment in the U.S. over the past 30 years. We show that investment is weak relative to measures of profitability and valuation — particularly Tobin’s Q, and that this weakness starts in the early 2000’s. There are two … [more]
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march 2016 by nhaliday

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