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Commerce · II.07 · MMXXVI · daylight

La Bourse  /  Volume II  /  Nº II.07  /  Workbook — the student

WORKBOOK — THE STUDENT

Chapter II.07 · Information, Signal, and Price

For the personal student. A term of practice, a term project, and a self-assessment. Everything here is applied to a life you are actually living, because the arithmetic in this chapter works at any scale and it is easiest to learn on decisions whose stakes you already understand.

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WHY THIS WORKBOOK IS DIFFERENT

Most courses teach you that prices convey information and leave it there as a sentence you agree with. This one asks you to compute how much, in bits, for prices you actually meet — and then to do something almost nobody does, which is to ask what the number was not pointed at.

By the end of a term you will have three things: a habit of counting bits before arguing about them, a working grasp of the value of information, and one small channel of your own that you built, ran, and either kept or retired on the evidence.

The tools are a calculator, a notebook, and lib/verify/II_07.py, which prints every input with its unit and its source so you can change one and watch the answer move. Change the inputs. That is the exercise. A number you have only read is a number you do not have.

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PART ONE — DISCOVERY

Weeks 1–4: count the bits in the prices around you

Exercise 1.1 — Your first entropy (90 minutes)

Pick a listed company you find interesting. Look up its share price, the tick size of the venue it trades on, and a daily return standard deviation over the last year. Then run the chapter's arithmetic by hand:

  sigma in currency   =  price x daily sigma
  effective support   =  sigma x 4.13273
  ticks spanned       =  support / tick
  bits                =  log2(ticks spanned)

At a hundred dollars, 2 percent and a cent, the answer is 9.691 bits. Write down what you get and, beside it, what you expected before you computed it. Most people expect a far larger number, and noticing the gap is the point of the exercise.

Exercise 1.2 — The shelf sweep (one week, ten minutes a day)

Choose five products you buy regularly. Photograph the price each day for a week. At the end, count how many changed.

Nakamura and Steinsson put the median monthly frequency of consumer price change at 8 to 11 percent excluding sales — H(0.09) = 0.4365 bits a month, 5.238 bits a year. Bils and Klenow, including sales, found a hazard near 21 percent — H(0.21) = 0.7415 a month, 8.898 a year. One alphanumeric character is log2(36) = 5.170 bits.

Now write the sentence that matters: everything the people who made this knew reaches me at roughly one character a year. Sit with it before deciding whether it is a complaint.

Exercise 1.3 — Find the channel that already works (2 hours)

Go and find one case, in your own life or your own field, where a number arriving somewhere new made things measurably better for everyone. Kerala's fishermen are the chapter's case — profits up about 8 percent, consumer prices down about 4 percent, waste to essentially zero, and nothing produced to achieve it.

Write half a page on yours. Name what moved, who gained, and what it cost to move it. You are building the appreciative habit: find the working thing first, then ask what it cannot do.

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PART TWO — THE ARITHMETIC

Weeks 5–8: the kernel, and why precision is the wrong lever

Exercise 2.1 — Write your own vector (2 hours)

Take one decision you make repeatedly — where to buy coffee, which flat to rent, which job offer to take, which supplier your side project uses. Write down every attribute that would genuinely change the decision. Not everything that matters: everything that would change it.

Count them. That is your k. Then write 1/k beside it — the share of a randomly oriented preference gradient a single price can be expected to recover. At k = 12 it is 8.3 percent. At k = 6, 16.7 percent. At k = 3, 33.3 percent.

Exercise 2.2 — Prove the kernel to yourself (45 minutes)

Take two options with identical prices. Write down the vector of attributes for each. Subtract. The difference vector is, by definition, in the kernel of the price functional — it is the thing the price is provably silent about.

Now do the part that makes it land. Ask: if the price were quoted to six decimal places instead of two, would any entry in that difference vector become visible? The answer is no, and the arithmetic says why:

  tick 0.01       ->   9.691 bits  ->  11 invisible dimensions
  tick 0.0001     ->  16.335 bits  ->  11
  tick 0.000001   ->  22.979 bits  ->  11

Write one paragraph explaining to a friend who has not read the chapter why an externality is a direction rather than a quantity. If you can write that paragraph, you have the chapter.

Exercise 2.3 — The cost of the channel (90 minutes)

Reproduce the chapter's division and then break it.

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

Now change one input at a time. At 3,000 companies it is 13,895 a bit; at 6,000, 6,948. At one-minute resolution the bandwidth rises 216.8 times and the cost falls to 48.07 a bit.

Then answer in writing: which of these numbers would you defend in a seminar, and which would you present with its boundary stated? The module names three boundaries at its head. Find them and copy them into your notes in your own words.

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PART THREE — DREAM AND DESIGN

Weeks 9–12: value the bit, then build one channel

Exercise 3.1 — Your first EVPI (90 minutes)

Take a real decision you are facing where you are genuinely uncertain. Estimate three numbers, honestly and roughly:

• p — the probability of the state that would change what you do
• L — what it costs you to act as if things are fine when they are not
• c — what it costs to take the safe action regardless

Then:

  without information        min(p L, c)
  with perfect information   p c
  EVPI                       min(p L, c) - p c
  bits supplied              H(p)
  value per bit              EVPI / H(p)

At p = 0.10, L = 1,000,000 and c = 50,000: pL = 100,000, c = 50,000, pc = 5,000, EVPI = 45,000, H(0.10) = 0.4690, value per bit 95,950.

Do yours. Then find where EVPI peaks by walking p: at p = 0.01 it is 9,500; at 0.05, 47,500; at 0.10, 45,000; at 0.30, 35,000; at 0.50, 25,000. It peaks at p = c/L and is zero at both ends. You cannot pay to learn something you already know, and you cannot pay to learn something that would not change anything.

Exercise 3.2 — Build one channel (four weeks)

Choose one attribute from your vector, one that is genuinely in doubt, and build a way of observing it. It can be small: a weekly log, a photograph, one question asked of one person, a count. The requirements are only these.

1. It is written down before you start — the baseline.
2. It is cheap enough that you will still be doing it in week four.
3. It is aimed at a decision that recurs.
4. You have said in advance what result would change what you do.

Point four is the whole exercise. A measurement with no pre-committed decision rule is a hobby, and you will find — most students do — that writing the rule is harder than collecting the data.

Exercise 3.3 — Design for the receiver (60 minutes)

Take your channel's output and compress it to four bits — sixteen possible messages. Then to one bit. Then ask which version you would actually read at the moment of decision.

Chile's front-of-pack octagon is a few bits and moved a national diet. The forty-number nutrition panel on the back of the same package had been there for years. Write one paragraph on why, using Sims's rational inattention: the binding constraint is the receiver's capacity, not the sender's.

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PART FOUR — DESTINY AND DELIGHT

Weeks 13–15: retire what you know, keep what you do not

Exercise 4.1 — The retirement review (60 minutes)

Look back at your channel. Has the answer stopped moving? If you have observed the same thing four times running, the posterior is no longer updating, H(p) is collapsing toward zero, and the measurement is now costing you attention for no decision change.

Retire it, and write down what you learned. Students find this the hardest instruction in the workbook and it is the most valuable. A bit spent on something you already know is a bit spent on nothing.

Exercise 4.2 — Find where the channel would lose (45 minutes)

Scale your channel down in your head. At what size does it stop being worth running? Use the chapter's form:

  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

The EU's own line is 25,000 tonnes under Article 27, and 1,000 tonnes for de minimis source streams. Write your own floor down. A proposal that does not know its floor gets defeated at the floor by someone who computed it first.

Exercise 4.3 — The pooling move (45 minutes)

If your channel is below its floor, do not abandon it — syndicate it. Find the threshold:

  members  >=  channel cost / EVPI per member
  20,000 / 1,500  ->  14 members, at EUR 1,429 each

Who are your fourteen? Name them. This is the exercise that most often turns a student project into something that outlives the term.

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THE TERM PROJECT

One piece of work, carried the whole way

The brief. Choose one decision that a group of people you belong to makes repeatedly — a household, a society, a team, a shared flat, a small business. Produce a document of about two thousand words containing:

1. The vector. Every attribute that would change the decision. State k and 1/k.
2. The entropy of the existing signal. Whatever number they currently use, computed in bits, with the inputs named.
3. The kernel. Which dimensions the existing signal is provably silent about, and the demonstration that precision would not recover them.
4. The EVPI table. Each candidate attribute, with p, L, c, EVPI, H(p) and value per bit. Ranked.
5. One channel, costed. What it costs a year, what it is worth a year, the break-even, and the pooling threshold if it does not clear alone.
6. A four-bit design. The actual signal, as the receiver would see it.
7. The retirement clause. What result would cause you to switch it off.

What makes a good one. Not ambition — honesty about the boundary. The best projects state clearly what their numbers did not look at. A project that computes one thing precisely and names three things it could not is worth more than one that estimates seven things and says so about none.

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SELF-ASSESSMENT

Score each honestly from one to five. Under three, go back to the exercise.

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CARRYING IT FORWARD

Three habits are worth more than the whole term if you keep them.

Count before you argue. Whenever somebody claims a signal carries something, ask how many bits and what the inputs are. The question is almost never rude and it is almost never asked.

Ask what the number was not pointed at. Not is this right — what is in the kernel. It is a different question and it produces different conversations.

Instrument the decision, not the world. A bit aimed at one decision was worth about 95,950 in the chapter's worked case, against 10,421 a bit for general price discovery. The ratio, 9.21, is a rule of thumb rather than a law — but the direction of it has never once been the other way.

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A NOTE ON READING THE SOURCES

Four papers carry this chapter and all four are short enough to read in an afternoon. Read them in this order, because the order is the argument.

Hayek (1945), eleven pages. Read it for the tin passage and for the phrase "a kind of symbol". Notice that every claim in it is an information claim made three years before the unit existed.

Shannon (1948). Read the first fifteen pages and stop. You need the definition of entropy and the idea of a channel with a capacity; the coding theorems are a different course.

Grossman & Stiglitz (1980), sixteen pages. Read it for the equilibrium condition and nothing else. The model is fully specified and you can follow the logic without following the algebra: informed traders must earn back their cost, so the price must stay partly uninformative.

Howard (1966), five pages. The shortest and the most immediately useful. It gives you the only defensible answer to should we measure this, and the answer is a number.

Then one you will have to find yourself. Every essay prompt in the assessment asks for a source the chapter does not cite. That instruction is not housekeeping: the habit of reading one thing nobody assigned is the difference between holding an argument and holding a position, and it is the single most transferable thing in this course.

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APPRECIATIVE QUESTIONS FOR YOUR SEMINAR

1. When has a single number told you something you could not have worked out any other way? What made it that good?
2. Where have you seen a small, narrow signal change behaviour that a large, rich one did not?
3. If your seminar could instrument one thing about its own work, what would be worth knowing that you currently guess?
4. What would you want to still be measured here in ten years, and what would have to be true for it to still be trusted?