La Bourse / Volume II / Nº II.07 / Ten concept briefs

Three colleagues at a whiteboard in a sunlit room, one writing while the others listen. — Plate II.07 · Ten concept briefsOne Number on a Chalkboard.All of that reaches the next person as one number. The chalk is not a summary of what she knows. It is the only part of it that can be transmitted.

TEN CONCEPT BRIEFS · Chapter II.07 — Information, Signal, and Price

One page each. A reader who reads only these ten pages has the chapter.

BRIEF 1 — A Price, Measured in Bits

The idea. A price is a message. Messages have sizes, and the size of this one can be computed from three things you can look up: the tick, the volatility and the horizon.

  daily sigma in currency   100.00 x 0.02   =  USD 2.00
  Gaussian constant         sqrt(2 pi e)    =  4.13273
  effective support         2.00 x 4.13273  =  USD 8.2655
  ticks spanned             8.2655 / 0.01   =  826.55
  entropy                   log2(826.55)    =  9.691 bits

Worked example. A hundred-dollar share, a 2 percent daily standard deviation, and the one-cent minimum increment SEC Rule 612 requires. The closing price carries 9.691 bits — one and a fifth bytes. The letter a in this sentence is eight bits.

And it barely moves. Take volatility from 0.5 percent to 4 percent — an eightfold change — and the message goes from 7.691 to 10.691 bits. Three bits for an eightfold move. Cut the tick from a cent to a millionth and you buy thirteen more, reaching 22.979. The size of a price is stubborn, which is both its great virtue and its hard ceiling.

Why it matters. Once you have a number, the conversation changes. You can divide by it, compare against it, and budget with it. Every argument in this chapter is downstream of somebody doing this division.

You already know this because you have watched a spreadsheet of forty columns get summarised into one score for a meeting, and you knew exactly what had been lost, and you could not have said how much.

BRIEF 2 — Hayek's Compression

The idea. The price system's achievement is that it throws almost everything away and still coordinates.

Hayek's 1945 argument is that the knowledge an economy runs on is dispersed, local, and often not articulable — the particular circumstances of time and place. No planner can gather it, because gathering it destroys it. What a price does is let each person act on their fragment while receiving, in one number, the net of everyone else's. He called it "a kind of symbol" in which "only the most essential information is passed on."

The worked case. Kerala, 1997–2001. Robert Jensen followed mobile coverage arriving in sardine fisheries. Before it, boats chose a beach before knowing the price there, and on some mornings eleven boats dumped a whole catch while buyers went unserved a short sail away. After coverage: dispersion collapsed, waste fell to essentially zero, fishermen's profits rose about 8 percent and consumer prices fell about 4 percent. Nothing was produced. One number moved earlier.

Why it matters. Any honest treatment of what prices cannot do has to start here, because this is the part that is not in dispute and not replaceable. A chapter that opened on the limits would be arguing with a thing it had not weighed.

You already know this because you have never once had to be told to economise on a material that got expensive.

BRIEF 3 — Shannon's Unit

The idea. Entropy is the average number of yes-or-no questions needed to identify an outcome. It is the unit Hayek needed and did not have.

  H(X) = - sum p(x) log2 p(x)

  binary case:  H(p) = -p log2 p - (1-p) log2(1-p)

Worked example — the shelf price. Nakamura and Steinsson put the median frequency of consumer price change at 8 to 11 percent a month excluding sales; Bils and Klenow found a monthly hazard near 21 percent including them.

  H(0.09) = 0.4365 bits/month  ->  5.238 bits/year
  H(0.21) = 0.7415 bits/month  ->  8.898 bits/year
  one alphanumeric character, log2(36)  =  5.170 bits

Two independent measurements, twenty years and two datasets apart, agree that a shelf price transmits between one and two characters a year. Everything the grower, shipper, processor and retailer know about how a thing was made reaches the person buying it at that rate.

Why it matters. Entropy is the honest denominator for every claim about what prices convey. Before it, the argument was aesthetic.

You already know this because you can tell how much a headline told you and how much it left out, and you have never needed a formula to feel the difference.

BRIEF 4 — The Kernel of a Price

The idea. A price is a scalar functional on a vector-valued world, so it has a null space — and the null space does not shrink when you add precision.

Let the state be x in R^k and the price be p = w · x. Two states differing by any vector in ker(w) produce the same price at any resolution, and I(p ; x⊥) = 0 is an identity.

  attributes k   channels   invisible dims   share invisible   gradient recovered
         12          1             11             91.7%              8.3%
         12          2             10             83.3%             16.7%
         12          3              9             75.0%             25.0%
         40          1             39             97.5%              2.5%

  precision instead, at k = 12:
  tick 0.01        ->   9.691 bits   ->  11 invisible dimensions
  tick 0.0001      ->  16.335 bits   ->  11
  tick 0.000001    ->  22.979 bits   ->  11

Three orders of magnitude of precision buys thirteen bits and moves the invisible dimension count by zero.

What this does to "externality." It stops being a quantity somebody got wrong and becomes a direction the instrument has no component in. You cannot reach it with better estimation, more liquidity, or more compute, because none of those change the rank of a linear map. Mount and Reiter proved the competitive message space is the smallest that clears a classical economy — minimality and blindness are one theorem read twice.

Why it matters. It tells you what to stop doing. Refining a price to capture a missing dimension is not a hard problem; it is the wrong problem.

BRIEF 5 — The Grossman–Stiglitz Impossibility

The idea. A fully informative price cannot exist in equilibrium.

If the price reveals everything the informed know, then nobody is paid for knowing it, so nobody pays to find out, so there is nothing for the price to reveal. The only consistent state is partial revelation: the informed earn back their research cost by trading with people who are not informed, and the market stays permanently, structurally, a little bit in the dark.

The two corollaries people miss.

Noise is load-bearing. Uninformed trading is not a defect in the market; it is the payroll of the research that makes the price worth reading. Fischer Black said it in one word in 1986 and the word was the title.
You cannot fix externalities by making prices more informative. The mechanism that would do it is the mechanism the result forbids.

Why it matters. It is the reason this chapter proposes other channels rather than better prices. The improvement route is closed by theorem, and knowing a route is closed is worth more than a decade of walking down it.

You already know this because you have worked somewhere that a piece of knowledge was valuable precisely because not everyone had it, and you understood without being told that publishing it would end that.

BRIEF 6 — The Price of the Price

The idea. The channel has a bill, and dividing the bill by the bits gives you a figure nobody quotes.

Kenneth French computed the cost of active investing in US equities: an average of 0.67 percent of aggregate market value a year from 1980 to 2006, and 101.8 billion dollars in 2006. Thomas Philippon found the unit cost of US financial intermediation near 1.87 percent of intermediated assets and essentially unchanged since 1886.

  4,000 companies x 252 days x 9.691 bits  =  9,768,479 bits/year
                                           =  1.164 MiB/year
  USD 101.8 bn / 9,768,479 bits            =  USD 10,421 per bit

The entire annual closing-price output of the US equity market is about one and a sixth mebibytes. At 6,000 listed companies the figure is 6,948 a bit; at 3,000, 13,895. The order of magnitude holds.

State the denominator. This looks only at closes. At one-minute bars the bandwidth is 2,101 bits a session — 216.8 times as much — and the cost falls to 48.07 a bit. But Grossman and Stiglitz already named what the extra is: the noise the arrangement requires. Bai, Philippon and Savov close it from the other side — measured aggregate price informativeness has not risen since 1960.

Why it matters. It makes "should we buy this measurement?" a comparable question rather than a values question.

BRIEF 7 — The Receiver's Channel Capacity

The idea. The binding constraint is not what the world can emit. It is what the decider can take in.

Christopher Sims's rational inattention starts from exactly Shannon's setup: an agent with a finite information-processing capacity chooses what to attend to, and the choosing is itself the economics. A person at a shelf has, generously, a few bits of attention. A committee has more, and nothing like a thousand datapoints' worth.

The worked case, and it is decisive. Chile's Ley 20.606 put a black octagon on the front of packages above thresholds for sugar, sodium, saturated fat and calories — a channel of a few bits, sitting beside the price. Taillie and colleagues measured a large fall in the volume of sugar-sweetened beverages households bought. A nutrition table carrying forty numbers had been on the back of the same package for years and had moved almost nothing.

The design rule that follows. Do not ask what should we disclose. Ask given four bits at the point of decision, which four? The good answers all look alike: narrow, unambiguous, and aimed at something genuinely in doubt.

Why it matters. It explains the whole pattern of what works and what gets repealed, and it explains it without anyone being at fault.

BRIEF 8 — The Second Channel, and Where It Loses

The idea. Missing dimensions need channels. Channels are metered, and below a scale they cost more than they are worth.

Take monitoring, reporting and verification at 20,000 euros a year per installation and carbon at 80 euros a tonne.

  500,000 tCO2e/yr  ->  EUR  0.04 per tonne of measurement
   25,000 tCO2e/yr  ->  EUR  0.80
    5,000 tCO2e/yr  ->  EUR  4.00
    1,000 tCO2e/yr  ->  EUR 20.00

  abatement needed to pay for measuring    20,000 / 80   =    250 tCO2e
  at a 10% abatement rate, the floor        250 / 0.10   =  2,500 tCO2e/yr

And the legislature agrees. Article 27 of Directive 2003/87/EC lets member states exclude installations under 25,000 tonnes — ten times this model's floor — and the monitoring regulation lets de minimis source streams under 1,000 tonnes use simplified methods. The people running the largest second price channel ever built have written into it that below a scale it is not worth running.

This is the honest negative of the whole chapter. The living-systems approach loses, cleanly, wherever measurement costs more than the decisions it changes are worth — which is most of the economy by count of enterprises.

Why it matters. A proposal that does not know its own floor will be defeated at the floor, by people who computed it first.

BRIEF 9 — The Value of a Bit

The idea. Information is worth exactly the decision change it produces, and nothing else.

Ronald Howard settled this in 1966. For a two-state, two-action problem with prior p on the state that would change the decision, a loss L from acting as if good when bad, and a remedy cost c:

  without information        min(p L, c)
  with perfect information   p c
  EVPI                       min(p L, c) - p c

Worked example. p = 0.10, L = 1,000,000, c = 50,000. Then pL is 100,000 and c is 50,000, so without information you buy the remedy at 50,000; with perfect information you buy it only in the bad state, at 5,000. EVPI = 45,000. Perfect information here supplies H(0.10) = 0.4690 bits, so the value per bit is 95,950 dollars — about 9.21 times the 10,421 a bit that general price discovery costs to produce.

The shape to carry. EVPI is zero at both ends and peaks where the decision is genuinely in doubt — at p = c/L, which here is 5 percent, where EVPI reaches 47,500. Measure what you cannot guess, and guess the rest. That is a theorem, not a temperament.

Why it matters. It separates important from worth measuring, which are constantly confused and are not the same thing at all.

BRIEF 10 — Pooling the Channel

The idea. A channel too expensive for one holder is affordable to fourteen.

  shared channel cost                      EUR 20,000 / yr
  EVPI at one small member                 EUR  1,500 / yr
  members before each one clears           20,000 / 1,500 -> 14
  cost per member at that size             EUR  1,429 / yr

The structure already exists in law. Regulation (EU) 2018/848 recognises the group of operators with an internal control system, precisely so that small holders can share one certification channel. The SO2 allowance market reached the same end by another route: it let the market allocate measurement and abatement effort to wherever it was cheapest, and Schmalensee and Stavins record compliance coming in far below the ex ante estimates, including the advocates' own.

The cheap channel survives. A thousand-dollar farm certification fee, against a USDA cost share of 75 percent capped at 750, leaves 250 dollars for one bit — one word on a label. At a 20 percent farmgate premium the break-even is 1,250 dollars of sales; unreimbursed, 5,000.

The expensive one does not. The Commission's Omnibus package of 26 February 2025 proposed removing roughly four-fifths of companies from sustainability reporting scope and stated a saving of 6.3 billion euros a year — about 157,500 per company, or 143 per datapoint across the eleven hundred in ESRS Set 1. That is the price of the missing dimensions, revealed by a repeal.

Why it matters. It is the whole strategy in one move: do not argue for the channel, syndicate it until it clears.

All figures in these briefs are computed in lib/verify/II_07.py, where every input is printed with its unit and its source, and the three boundaries of the computation are stated at the head of the module.