La Bourse / Volume II / Nº II.07 / Quiz, reflection, essays

Three colleagues at a whiteboard in a sunlit room, one writing while the others listen. — Plate II.07 · Quiz, reflection, essaysOne 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.

ASSESSMENT · Chapter II.07 — Information, Signal, and Price

Three instruments: a ten-point quiz, eight reflection questions, five essay prompts. The quiz checks comprehension rather than recall. The reflections are private and first-person. The essays are arguable from more than one side.

THE QUIZ — ten points

Four on recall.

1. Write Shannon's binary entropy and say what it measures.

H(p) = -p log2 p - (1-p) log2(1-p), in bits. It is the average number of yes-or-no questions needed to identify the outcome of a two-state variable. One mark for the expression, one for saying that H(0) and H(1) are both zero — a certainty carries no information, which is the whole basis of the value-of- information argument later in the chapter.

2. State the Grossman–Stiglitz impossibility result in two sentences.

If prices fully reveal what informed traders know, nobody is compensated for becoming informed, so nobody does, so prices reveal nothing. Equilibrium therefore requires prices to be only partially revealing, with the informed paid out of trade with the uninformed.

3. A price is a scalar functional p = w · x on a state in R^k. What is the dimension of its kernel, and what happens to that dimension when the tick size falls?

k − 1, and it does not change. Precision buys bits and buys no dimensions. At k = 12 the price leaves eleven dimensions — 91.7 percent — exactly indistinguishable at any resolution.

4. Name the two thresholds the EU ETS itself writes into law for when a second channel is not worth running.

Article 27 of Directive 2003/87/EC: member states may exclude installations under 25,000 tCO2e a year. Implementing Regulation (EU) 2018/2066: de minimis source streams under 1,000 tCO2e a year may use simplified methods.

Four on application.

5. A colleague proposes that better data science will let market prices internalise a supply chain's labour conditions. Diagnose the proposal.

It confuses precision with dimension. Labour conditions lie in the kernel of the pricing functional; adding resolution moves the bit count and not the rank. Full marks require both halves: naming the null-space argument, and naming Grossman–Stiglitz as the second, independent reason the improvement route is closed — a price that fully revealed its inputs would pay nobody to produce them.

6. Two suppliers quote an identical price for an identical specification. What, precisely, do you now know about them, and what do you not?

You know they are price-identical, which means they differ — if at all — by a vector in the kernel. You know nothing about the eleven other dimensions, and critically you do not know that they are similar on them; price-identity carries no information about the orthogonal complement at all. Credit any answer that names the next move: pick the one or two dimensions that would change the decision and instrument those.

7. Chile's warning label carries a few bits. A nutrition panel on the same package carries far more and moved almost nothing. Explain, using the chapter's framework, why the smaller channel did more.

The binding constraint is the receiver's processing capacity, not the sender's bandwidth — Sims's rational inattention. A channel wider than the receiver can read delivers zero, regardless of its contents. The stronger answer notes that this also predicts the failure of thousand-datapoint reporting regimes, and that it is a design rule rather than a criticism of anyone.

8. Why does the chapter say a regulator-funded measurement regime will degrade while a customer-funded one will not?

Because no party inside a compliance-funded channel earns anything from its being accurate. Grossman–Stiglitz generalised: the accuracy of a channel is sustained by somebody's income depending on it. A customer paying for the attribute supplies that party; a filing requirement does not.

Two that require the arithmetic to be done.

9. A stock trades at 50 dollars with a daily return standard deviation of 3 percent, quoted in cents. How many bits does its closing price carry, and how many more would you gain by moving to a tenth-cent tick? Show your working.

sigma = 50 × 0.03 = 1.50. Effective support = 1.50 × sqrt(2πe) = 1.50 × 4.13273 = 6.199. Ticks spanned = 6.199 / 0.01 = 619.9. log2(619.9) = 9.28 bits. A tenth-cent tick is a factor of ten, so log2(10) = 3.32 more bits, giving 12.60. The mark is for noticing that a tenfold precision gain buys 3.32 bits and zero dimensions — the point of the whole chapter, arrived at by arithmetic rather than assertion.

10. An attribute has a 20 percent prior on the state that would change your decision. Acting as if good when it is bad costs 400,000. The remedy costs 50,000 a year. A measurement channel costs 18,000 a year. Should you buy it, and what is the value per bit?

pL = 0.20 × 400,000 = 80,000; c = 50,000; so without information you buy the remedy at 50,000. With perfect information you buy it only in the bad state: pc = 0.20 × 50,000 = 10,000. EVPI = 50,000 − 10,000 = 40,000. Against an 18,000 channel, buy it — and note it clears by better than two to one. H(0.20) = 0.7219 bits, so value per bit = 40,000 / 0.7219 = 55,407. The stronger answer observes that EVPI here is bounded above by the remedy cost, so no measurement of this attribute can ever be worth more than 50,000 a year, which caps the channel budget before anyone negotiates.

REFLECTION — eight questions, for one person and a pen

These are not for a room. Write the answers by hand if you can; the slowness is the point.

Think of something you know about your own work that has never reached a price, a metric or a report. What is it, and who would act differently if it had?

Where are you currently spending attention on a measurement whose answer you already know? What would you do with that attention if you retired it?

Recall a decision you made well on very little information. What were you reading that was not written down?

Which of the dimensions you care about most is one you have never tried to measure because it seemed unmeasurable — and is that true, or is it that no decision depended on it?

When have you mistaken important for worth measuring? What did that cost, and how long did it take to notice?

Think of a number you produce that somebody else relies on. Whose income depends on it being accurate? If nobody's, how do you keep it honest, and is that sustainable after you leave?

Where in your life are you treating one number as though it carried everything — a salary, a valuation, a score, a weight? What sits in its kernel?

What is the smallest honest signal you could start sending that somebody downstream of you would actually read? Write it in fewer than ten words.

ESSAY PROMPTS — five

Each is arguable from more than one side. Each requires at least one source the chapter cites and at least one it does not.

1. Hayek was right and it does not help. The chapter accepts Hayek's compression argument entirely and then argues that the same property that makes the price system work makes it structurally unable to carry multi-dimensional constraints. Argue either that this is a genuine tension the socialist calculation debate never addressed, or that Hayek's position already contains the answer — that non-price institutions were always meant to carry the rest. Use Hayek (1945) and Mount & Reiter (1974), and one source on the calculation debate the chapter does not cite.

2. Is the null-space argument doing real work? The chapter's central move models a price as a linear functional and concludes that k − 1 dimensions are invisible. Argue against it: prices are not linear in attributes, markets support many prices across many goods, and a system of prices may span more dimensions than any one of them. Then give the strongest defence. Engage Mount & Reiter or Jordan (1982) directly, and at least one source on general equilibrium's informational requirements that the chapter does not cite.

3. The noise that pays for the signal. Grossman and Stiglitz require uninformed trade for informed trade to be worth doing. Argue either that this makes a large share of financial activity a socially necessary cost of price discovery, or that it makes it a transfer dressed as a service. Use Grossman & Stiglitz (1980) and French (2008), and one empirical source on trading costs or market-maker profits that the chapter does not cite.

4. The repeal as evidence. The chapter treats the European Commission's 2025 Omnibus package — and its stated 6.3 billion euro annual saving — as a revealed price for the missing dimensions. Argue that this is a legitimate measurement of what the bandwidth cost, or that it is a measurement of political appetite that tells us nothing about cost. Use the chapter's material on ESRS and Schmalensee & Stavins (2013) on the SO2 programme's ex ante cost estimates, and one source on regulatory cost-benefit estimation that the chapter does not cite.

5. Narrow channels and what they crowd out. Chile's octagon worked because it was small; the chapter makes narrowness a design rule. Argue the counter-case: that a few-bit signal necessarily collapses a rich judgement into a binary, that the attributes chosen become the attributes optimised, and that the crowding-out may cost more than the signal gains. Use Taillie et al. (2020) and Ostrom (1990) on monitoring design, and one source on Goodhart's law, measurement gaming or indicator crowding-out that the chapter does not cite.