La Bourse / Volume VII / Nº VII.08 / Quiz, reflection, essays

A watercolour of open fields running to distant hills, wildflowers in the grass. — Plate VII.08 · Quiz, reflection, essaysThe Marker Field.These are not headstones. They are a message to somebody ten thousand years from now, written in geometry because no language is expected to survive. Somebody costed them, approved them and signed for them — which means somebody put a number on the year twelve thousand.

ASSESSMENT · Chapter VII.08 — The Long View and the Discount Rate

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. State Weitzman's result in one sentence, and say what it depends on.

With a fixed but unknown discount rate, the certainty-equivalent rate declines with the horizon toward the lowest rate in the support. One mark for the statement; one for naming the dependency — the limit is the lowest rate you admit as possible, so the result is exactly as defensible as the floor somebody chose. It is a theorem about averaging, not an ethical preference.

2. Why must you average discount factors rather than rates?

Because the factor is what a pound is worth and the rate is only its logarithm. E[e^-rt] is not e^-E[r]t. Averaging rates discards the fact that low-rate branches retain almost all of the surviving value at long horizons.

3. What did Ramsey himself say about pure time preference, and what technical problem does his position create?

That discounting future utilities is "ethically indefensible and arises merely from the weakness of the imagination." With delta = 0 an infinite horizon has no finite total to maximise; Ramsey's own device was the Bliss point. Koopmans and Diamond later proved no complete, continuous, anonymous and Pareto ordering of infinite streams exists.

4. What is the Yucca Mountain standard now, and what instrument does it use?

One million years, following NRDC v. EPA (D.C. Cir. 2004) and EPA's 2008 final rule: 15 millirem a year to year 10,000, 100 millirem median thereafter. The instrument is a dose ceiling — a constraint on the state of the world — not a discount rate.

Four on application.

5. A colleague argues: "We know nothing about the year 2300, so it is unserious to weight it." Answer them with the chapter's arithmetic.

Widening the rate distribution while holding the mean at four percent raises the present value of a year-300 benefit by up to 12,106 times. The uncertainty is the argument for weighting it, because low-rate branches survive an average that high-rate branches have annihilated themselves out of. Full marks require naming that the mean is held fixed — the effect is the spread alone.

6. Your treasury adopts the Green Book declining schedule for a 200-year commitment. Your successor thirty years from now reprices it downward by a third and says nothing improper has occurred. Are they right?

Yes. The published table is indexed by years-from-now and re-bases with every present, so applying it correctly at year 30 gives 64.5 percent of the plan's own continuation value. The stronger answer distinguishes this from the theorem's schedule, which is indexed to calendar date and to what has been learned and is time-consistent — and notes that no finance ministry publishes that form.

7. A board paper says a decommissioning duty is "forty percent funded." What is the one question that makes that number mean something?

At what discount rate. A funding ratio is a discount rate wearing a percentage sign. The usable form is the inverse: the real return the pot implicitly asserts against the undiscounted schedule, which can be compared to the fund's own realised return. Credit any answer that also names the horizon.

8. A climate paper argues for a low discount rate on total-utilitarian grounds, citing population growth. Under what condition does that argument reverse, and when does the condition arrive?

When population growth n turns negative, since the total criterion gives delta - n + eta·g. On UN medium projections world population peaks in the 2080s; on the low variant n is already −0.208 percent a year to 2100, which raises the rate to 3.71 percent and lowers the future's weight to 0.86 of the average-criterion value.

Two that require the arithmetic to be done.

9. Three rates — 1, 4 and 7 percent — are equally likely and persistent. Compute the certainty-equivalent spot rate at year 300 and the present value of one million pounds arriving then. Show your working.

E[e^-r·300] = (e^-3 + e^-12 + e^-21)/3. The terms are 0.0498, 6.144e-06 and 7.583e-10. The sum is 0.049793; divided by three, 0.01659774. PV = £16,597.74. Spot rate = -ln(0.01659774)/300 = 4.0985/300 = 1.3662 percent. The point of the question is the second line of the working: the 7 percent branch contributes seven parts in ten billion and is arithmetically irrelevant. Credit any method reaching 1.36–1.37 percent and £16,000–£17,000.

10. A duty of £130bn falls evenly across 120 years. A trust holds £15bn. What real return is that trust asserting, and what would it need to hold at a realised 4 percent real?

Level liability 130/120 = £1.0833bn a year. Solve 1.0833 · [(1 − (1+r)^-120)/r] = 15 for r: the annuity factor needed is 15/1.0833 = 13.846, giving r = 7.22 percent real, forever. At 4 percent the annuity factor is 24.774, so the required pot is 1.0833 × 24.774 = £26.84bn. The stronger answer states the conclusion in the right register: the trust is not "twelve percent funded", it is carrying an unstated forecast of 7.22 percent real in perpetuity against a fund realising four.

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.

What is the longest horizon you have ever personally committed to — in work, in money, in a relationship, in a promise — and what made you able to hold it?

Name something you inherited that somebody paid for and never saw used. What do you actually feel toward them? Write the sentence you would say to them.

Where in your own life are you running a horizon-indexed plan that re-bases every time you look at it — always five years out, never arriving?

Ramsey called discounting future people a weakness of the imagination. Where is your imagination weak about your own future? Be specific about the year.

What have you refused to price because pricing it felt like a category error? Was that instinct right, and how would you know?

Think of a commitment you made that a later version of you wanted to undo. Did you want to undo it because you had learned something, or because you had arrived at the present? Those are different, and you know which it was.

If you had to leave one constraint behind you — a standard rather than a sum, a state of the world rather than a payment — what would it be?

Who will be alive in a hundred years because of a decision somebody makes this decade? You cannot name them, and that is the whole difficulty. Write what you owe them anyway, in one sentence, and see whether it survives the reading.

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. Does the non-identity problem weaken long-horizon obligation, or relocate it? The chapter argues it defeats claims of individual harm while leaving claims about the quality of the world standing, and that this settles the choice of instrument in favour of standards over net present value. Argue the counter-case: that impersonal criteria import the Repugnant Conclusion and that a person-affecting view, properly wide, can survive. Use Parfit (1984 and 2017), and one source in population ethics the chapter does not cite.

2. Is a declining schedule defensible, given that it is time-inconsistent? The chapter concedes that a successor re-applying a horizon-indexed table is correct by its own lights, and measures the gap at 64.5 percent. Argue either that the calendar-form schedule solves this and ministries should publish it, or that the inconsistency is fatal and a constant low rate is the honest instrument. Use Weitzman (1998) and Gollier and Weitzman (2010), and one source on fiscal rules or constitutional political economy the chapter does not cite.

3. The Nuclear Waste Fund collected faithfully and the repository was not built. Write this as an argument against funded trusts as commitment devices — the money was a substitute for the action, and the appearance of solution delayed the solution. Then write the strongest rebuttal. Conclude with which you find more persuasive. Use the Nuclear Waste Policy Act and NARUC v. DOE (2013), and one account of the Yucca Mountain siting process the chapter does not cite.

4. Sovereign wealth funds and the per-head condition. Norway's rule clears its per-head sustainability line by 0.40 percentage points; Alaska's five percent draw against a CPI-plus-five objective leaves zero real growth, so the fund shrinks per head at the population growth rate. Argue whether a constitutional entrenchment that protects principal but not share per head is a genuine intergenerational commitment or a well-designed illusion. Use the Alaska Constitution Article IX section 15 and NBIM's reporting, and one source on Dutch disease or permanent-income rules the chapter does not cite.

5. Should the far future be priced at all? The chapter shows a pound in year 300,000 is worth ten to the minus 1,296 at the Green Book's own floor, and argues that past a few thousand years a constraint is the correct instrument. Argue the opposite: that abandoning pricing removes the discipline of trade-offs and licenses unlimited present cost for unmeasurable future benefit, and that a very low rate is preferable to no rate. Use the EPA's 2008 Yucca rule and Weitzman (2009) on fat tails, and one critique of precautionary or constraint-based regulation the chapter does not cite.