Skill, Mastery, and Time

Volume V — Labour, Value, Flourishing

Nine movements, one apprentice.

THE PLATE

A woman reading on a daybed, the shadows of palm leaves falling across the wall behind her in late sun. — Plate V.05The Twenty-Eighth Month.There is a month in every apprenticeship when the firm stops paying for the learning and starts being paid by it. In Switzerland it is month twenty-eight of thirty-six, and almost nobody — including the apprentice — knows the date.

THE LETTER

You have almost certainly heard someone say, in a meeting about training, that the problem is people leave. It is said in a particular tone: regretful, worldly, as though naming a law of nature. And it ends the conversation, which is what it is for.

This chapter is about the arithmetic underneath that sentence, because the arithmetic turns out to be far more interesting than the sentence. People do leave. In Switzerland more than sixty per cent of apprentices leave the firm that trained them within a year of qualifying — a higher rate of loss than anywhere else in Europe — and Swiss firms train more deeply and more willingly than almost anyone. Those two facts sit together comfortably, and understanding why is the whole of this chapter.

The short version is that skill is a capital asset with a depreciation rate, and nobody capitalises it. Not the firm, whose accounts show training as an expense in the period it was incurred. Not the worker, who has no balance sheet. Not the tax authority, which allows the deduction and asks no further questions. A factory that treated a lathe this way — expensed on purchase, tracked nowhere, depreciated at a rate nobody had measured, and then described as "lost" when it walked out of the building — would be regarded as badly run.

What follows is the asset, priced. The returns to training that the credible literature actually supports. The published cost-benefit surveys that tell you, to the month, when a trainee stops costing and starts paying. The depreciation rate, which is not one number but a range covering a factor of eight, and which decides everything downstream. The poaching externality, measured rather than lamented. And then the four instruments that make training rational for a single firm when, left alone, it is not.

You will not need to believe anything about human potential to use this chapter. The case is financial and it holds on its own.

— The Editors

DISCOVERY

What is already working

Switzerland trains its young people at a profit, and has the receipts.

Since 2000 the Swiss have run four national cost-benefit surveys of apprenticeship training, each a representative sample of firms answering detailed questions about what an apprentice costs and what an apprentice produces. The third, by Strupler and Wolter in 2012, covered 2,352 firms. The fourth, for the 2016/17 training year, covered more than 5,700 training firms and about 4,000 firms that do not train — which matters, because a survey of only the trainers would be a survey of the persuaded.

The headline is one line long. Gross costs of CHF 28,070 per apprentice-year. Productive output of CHF 31,240. A net benefit of CHF 3,170 per apprentice per year, before the firm has hired anybody. Around two-thirds of Swiss training firms end an apprenticeship ahead; the remaining third carries a net cost and recovers it, if at all, by keeping the person.

This is not a subsidy story and it is not a moral story. It is a story about where the payback sits in time, and we will compute the date in the next movement.

Germany runs the other design, deliberately, and it also works.

The German surveys, run by BIBB, show the mirror image. In the 2007 wave (Schönfeld and colleagues, 2,986 interviews) the average firm carried a net cost of €3,596 per apprentice-year — down by around a third from the 2000 figure of €6,033, because apprentices were being used far more in real production. Seventy per cent of German firms still finish an apprenticeship out of pocket. They recover it afterwards: slightly more than half of German apprentices are still with the training firm a year after qualifying, against roughly a third in Switzerland.

Two countries, adjacent, sharing a language and a manufacturing tradition, have solved the same problem in opposite directions — and both solutions are stable. That is the appreciative finding, and it is worth more than either country's model taken alone, because it tells you the design space has at least two good points in it rather than one orthodoxy.

Austria holds the third position. Of 383,336 people who finished an apprenticeship period between 2011 and 2020, about three-quarters passed the final examination; around seventeen per cent left early. Austrian firms historically carried net costs, like German ones, with roughly a third generating a net benefit during training.

And the theory caught up with the practice. Gary Becker's 1964 result was that in a competitive labour market no firm will pay for general skills, because the trained worker can sell them anywhere; the worker must therefore pay, by accepting low wages while training. The German apprenticeship system is a standing contradiction of this, and for thirty years that was treated as a curiosity.

Daron Acemoglu and Jörn-Steffen Pischke resolved it. In Why Do Firms Train? (1998) and Beyond Becker (1999) they showed that Becker's conclusion depends entirely on the wage rising one-for-one with productivity. Where the wage structure is compressed — where a worker who becomes twenty per cent more productive gets ten per cent more pay — the firm keeps the difference, and the difference is a return on the training. Compression is produced by ordinary things: search frictions, collective agreements, minimum wages, and above all the training firm's private knowledge of how good this particular person actually is. Technologically general skills become economically specific, because the outside market cannot see them.

Read that carefully and notice what it licenses. It means firm-financed general training is not charity and not an anomaly. It is the rational response to a particular wage structure, and a wage structure is something an institution can choose.

The last piece of what is working: the value of a year. Psacharopoulos and Patrinos assembled 1,120 estimates from 139 countries and found the private return to an additional year of schooling to be about nine per cent a year, stable across decades, higher for women and higher in poorer countries. Nine per cent, compounding, on an asset nobody insures.

THE ARITHMETIC

What works, what does not, and where the line sits

First, why a Mincer coefficient can be read as a rate of return at all.

Take a nine per cent return on a €40,000 wage: €3,600 a year, for a working life of forty years. Discounted at five per cent real, the annuity factor is 17.159, so the present value is €61,773 against a cost of one year's forgone earnings. Net present value €21,773. Solve for the rate that sets the two equal and the internal rate of return is 8.68 per cent — 0.32 of a point below the Mincer coefficient itself. Over a long career the coefficient and the IRR converge, which is exactly why labour economists treat them as the same object, and why they stop doing so for short courses.

Second, the number this chapter turns on: the depreciation rate.

Ask three good studies and you get three different answers, and the difference between them is not noise — it is the whole design problem.

  route                              delta      half-life   what it measures
  ------------------------------------------------------------------------
  Arrazola & de Hevia (2004)         1.35 %/yr    51.3 yr   general human capital
  Hanushek et al. (2017)             3.04 %/yr    22.8 yr   vocational advantage
  Deming & Noray (2020)             10.41 %/yr     6.7 yr   applied STEM skill

The last is computed from their own published figures: the applied-STEM earnings premium is 44 per cent at age 24 and 14 per cent at age 35, so ln(0.44 / 0.14) / 11 = 0.1041. At that rate 64.7 per cent of the premium is gone within ten years, which is what they report — "declines by more than 50 per cent in the first decade of working life."

The middle row is computed the same way. Across eleven countries, men with a general education are 6.9 percentage points less likely to be employed at labour-market entry than men with a vocational education, and that gap closes by 2.1 points every ten years. Divide: the advantage is exhausted at age 16 + 6.9 / 0.21 = 48.9, and Hanushek and his colleagues report the crossover at forty-nine. In the German Microcensus the turn comes at forty-three, and by sixty-five the general-education man is 11.5 points more likely to be employed.

So: from 1.4 per cent a year to 10.4 per cent a year. A factor of eight. There is no such thing as the depreciation rate of a skill. There is the depreciation rate of that skill, and every instrument below is chosen by it.

Third, the Swiss apprenticeship, to the month.

Strupler and Wolter published the components of an apprentice's productive value by year: time actually spent in the firm, share of that time spent on productive work, and productivity relative to a qualified worker. For a three-year Swiss cook those are 0.79 × 0.77 × 0.40 in year one, rising to 0.81 × 0.84 × 0.80 in year three — relative productive values of 0.26, 0.41 and 0.54.

Spread the survey's CHF 93,720 of three-year output across those shares, hold costs at the survey average of CHF 28,070, and the ledger reads:

  year    output      cost        net     cumulative
  --------------------------------------------------
    1     20,138    28,070     -7,932        -7,932
    2     31,756    28,070     +3,686        -4,245
    3     41,825    28,070    +13,755        +9,510

The cumulative line crosses zero at month 27.7 of thirty-six. Let the costs rise across the years instead of sitting flat — apprentice pay climbs, instructor time falls — and it crosses at month 26.2. Under both profiles the Swiss firm is ahead somewhere between the twenty-sixth and the twenty-eighth month, eight or nine months before the apprentice is free to leave. Discounted at five per cent the three years show a net present value of CHF 7,861.

Fourth, the honest negative. The German model is barely solvent.

Three years at €3,596 is a net investment of €10,788, recovered only after qualification and only from people who stay. At the measured German retention of 53 per cent, recovering it over five years requires holding back €4,071 a year — about nine per cent of a skilled worker's pay, every year, for five years. That is the Acemoglu-Pischke rent, priced.

Now ask how much slack there is. At a rent of €4,000 a year over five years the break-even retention rate is 53.9 per cent. Measured German retention is 53 per cent. The model clears by less than a point, and it clears only because German institutions hold mobility down. Run the identical firm at Swiss mobility — 35 per cent retention — and the required rent becomes €6,165 a year, 13.7 per cent of skilled pay, which no wage structure in Europe delivers. The German design is not portable. It is the local solution to a local labour market, and exporting it to a fluid one produces a training system that quietly stops.

Fifth, and this is the cut. Set a rent of €4,000 a year against each of the three depreciation rates, at German retention, over five years:

  delta    five-year value   cover of the 10,788 investment
  -----------------------------------------------------------
   1.35%      10,250 EUR                 0.95x
   3.04%       9,833 EUR                 0.91x
  10.41%       8,264 EUR                 0.77x

Over an unbounded horizon the same three rents are worth €157,037, €69,657 and €20,364 — a ratio of 7.7 between the slowest and the fastest.

The instinct in the room is skills change too fast to be worth investing in. The arithmetic says something different and sharper. Rapid obsolescence does not destroy the case for training. It destroys the case for a long payback — and those are different sentences with opposite consequences. A skill with a 6.7-year half-life cannot support a design that recovers its cost over eight years of retained tenure. It can support a design that recovers the cost by month twenty-eight of thirty-six, while the person is still under contract, and is then indifferent to whether they stay.

Which is to say: the faster skills go stale, the more the world needs the Swiss design and the less it can afford the German one. The poaching problem is not solved by holding the worker. It is dissolved by moving the payback in front of the worker's freedom to leave.

DREAM

What becomes ordinary

In the firm that has understood this, the training budget has stopped being a line in the discretionary column and has become a schedule with a shape.

Every training investment carries three numbers on the page where it is approved: what it costs, how fast the skill it builds depreciates, and the month at which cumulative productive contribution overtakes cumulative cost. The third number has a name — the payback month — and everyone in the building uses it, including the people being trained, who are told theirs on the day they start. It is not treated as a secret and there is nothing embarrassing in it. It is the same courtesy as telling someone what their machine cost.

Learning happens in production rather than beside it, because that is what moves the payback month earlier, and because the survey evidence says skill acquisition and productive work are complements rather than rivals. The apprentice is not simulating the job in a training room for eighteen months. She is doing a graded slice of the real job from the first week, and the slice grows on a schedule somebody designed.

The depreciation rate is a live input, not an opinion. The firm knows which of its skill families sit near one per cent a year and which sit near ten, because it looks — at how fast its own job descriptions change, at how quickly a returning parent needs re-grounding, at what a five-year-old certification is worth in a live problem. Fast-depreciating families get short, dense, frequently-repeated training with the payback inside the engagement. Slow-depreciating families get deep, long, expensive formation, and the firm is relaxed about a seven-year horizon because the asset is still there in seven years.

Nobody argues about poaching, because the word has stopped describing anything. A firm whose payback lands inside the contract term has no exposure to departure. A firm whose payback lands outside it either shortens the payback or joins a pool, and both are ordinary commercial decisions taken in a Tuesday meeting.

And the person who leaves is not a loss that somebody has to absorb quietly. She is a graduate of a programme that has already paid for itself, carrying a skill the firm can prove she has, into a labour market where the proof travels. The firm that trained her is, quite openly, better at being left than its competitors — and that turns out to be why the good ones come to it.

DESIGN

The four instruments, and which one your numbers choose

Four mechanisms make training rational for a single firm when, unaided, it is not. They are not alternatives on a menu; each has a jurisdiction where it is known to work and a condition under which it fails.

One — move the payback inside the contract. (Switzerland.)

The strongest instrument is not financial at all. It is curriculum design. The Swiss payback lands at month 27.7 because apprentices spend roughly four days a week in the firm, most of that time on productive work, at a relative productivity that reaches 80 per cent of a qualified worker by year three. Every one of those three numbers is a design choice written into a national training ordinance by the professional association for that occupation, with firms in the room.

The condition it needs: an occupational standard specific enough that a qualification means the same thing in every firm, and employer bodies with the authority to write it. Without the standard, the apprentice's productivity is unverifiable and the payback cannot be relied upon.

Two — levy-grant. (The United Kingdom, since April 2017.)

Employers with a pay bill above £3 million pay 0.5 per cent of the excess into a digital account, with a £15,000 allowance, a ten per cent government top-up, and a twenty-four-month expiry on unspent funds. A firm with a pay bill of eight million pounds pays £25,000 and holds £27,500 to spend.

The measured outcome, honestly. National starts fell from 494,900 in 2016/17 to 375,800 in 2017/18, down 24.1 per cent. But the careful evaluation — Patrignani, Conlon, Dickerson and McIntosh, matching levy payers to similar non-payers — found training intensity rose at levy-paying firms by between 0.1 and 0.5 percentage points, with no displacement of other training. Both facts hold. The levy raised training where it landed; the national number fell because three other reforms bit at the same moment.

At 229 employees and a 0.5-point effect, that £25,000 buys 1.14 additional starts — £21,875 per marginal start, and £109,375 at the low end of the estimate, because most of the money funds apprentices the firm was going to train anyway. A levy is a price signal, not a programme, and it should be costed as one.

Three — the training bond, and its ceiling. (Germany, the United Kingdom, the United States.)

The obvious answer is to make the worker repay if they leave early, and every jurisdiction that has examined it has capped it hard. German labour-court doctrine ties the permitted lock-in to the length of the training: a month of training supports about six months, two months supports a year, three or four months supports two years, and the repayment must taper month by month. After Cavendish Square v Makdessi (2015) an English court asks whether the sum is proportionate to a legitimate interest; a figure not grounded in evidenced cost is a penalty and is void. In the United States, training repayment agreements have drawn regulatory attention precisely as employer-driven debt.

And in the case where the money is largest, German law removes the instrument altogether: §12(1) of the Berufsbildungsgesetz voids any clause restricting an apprentice's occupational activity after training ends, with one narrow exception for a commitment to employment agreed in the final six months. The €10,788 investment cannot be bonded at all.

The general rule is worth carrying: a bond big enough to matter is unenforceable, and a bond the law will keep is too small to matter.

Four — collective provision. (Denmark, since 1977.)

Every Danish employer, public and private, pays a flat per-head contribution to the Employers' Reimbursement Fund — DKK 2,821 per full-time employee at the 2026 rate, so DKK 564,200 a year for a 200-person firm, about €75,600. Firms that host apprentices draw reimbursement for apprentice wages during school periods, and there is an annual quota with a bonus for exceeding it and a surcharge for missing it.

The design is the cleanest statement of the principle in Europe: the free-rider and the trainer pay exactly the same, and only the trainer is reimbursed. The externality is not appealed to. It is priced.

DESTINY

How it holds when nobody is pushing

A training system sustains itself when the firm's private arithmetic points the same way as the public interest, and it decays quietly whenever those two drift apart. The Swiss system holds because each firm, acting selfishly, is ahead by month twenty-eight. Nothing needs to be enforced. The German system holds because wage compression, works councils and sectoral agreements keep mobility low enough for a five-year recovery — and its solvency margin, computed above, is under one percentage point of retention.

Name the failure modes plainly, because they are visible before they are fatal.

Contract dissolution is the first. German premature dissolutions rose from 25.0 per cent in 2013 to 29.5 per cent in 2022. Swiss terminations run at 20 to 25 per cent — though between half and three-quarters of those learners resume training within two or three years, which is a different fact and should not be counted as the same one. A dissolution in year one destroys the entire investment; a dissolution in month thirty destroys nothing, because the payback has already landed. This is the second reason the payback month matters, and it is the one firms notice first.

The life-cycle reversal is the second, and it is the honest negative that does not go away. The vocational employment advantage is 6.9 points at entry and zero by age forty-nine, and negative by eleven and a half points at sixty-five. An apprenticeship system that trains magnificently at eighteen and offers nothing at forty-five has not solved the problem. It has moved it to a place where nobody measures it. A system that holds is one where the same institution that trains apprentices also trains the forty-five-year-old, on the same standard, with the same instruments — and almost none of them do this yet.

The third is quieter. When mobility rises, the German design fails before anyone announces it: firms do not declare that they have stopped training, they simply take one fewer apprentice a year, and the sector notices seven years later.

DELIGHT

What it feels like

There is a moment in any real apprenticeship — trades, surgery, code, kitchens — when the person stops checking whether they have done it right and simply knows. It arrives without ceremony, usually in the middle of something ordinary, and it is almost always noticed by the teacher before the learner.

What this chapter adds is that the moment has a shadow in the accounts. Somewhere near the same week, the cumulative line crosses zero. The firm stops paying for the learning and starts being paid by it, and the two events are the same event seen from two sides of the bench.

There is a particular pleasure in knowing that. It converts a relationship people usually describe with slight embarrassment — we're investing in her, said as though it were generosity — into something cleaner and more equal. She is not being carried. She is, from month twenty-eight, carrying. And the older hand who stopped watching did that because the work was right, not because they were being kind.

The best-run workshops in Europe have known this in their hands for four hundred years. It is pleasant, and slightly funny, that the surveys agree.

OPERATIONALIZE THIS

At the level of finance

The instrument: a group training organisation with a reimbursement pool.

A training consortium — a Lehrbetriebsverbund in Switzerland, an Ausbildungsverbund in Germany, a Group Training Organisation in Australia — is a legal entity that employs apprentices and places them with member firms for rotations. It is the right instrument for exactly the firm this chapter is about: one whose payback month falls after the tenure it can count on, and which therefore cannot train alone.

The structure. The consortium is the employer of record and holds the training contract. Member firms host, pay a hosting fee per apprentice-month, and pay an annual subscription per employee whether or not they host — the Danish principle, implemented privately. Hosting fees and subscriptions together fund apprentice wages, the training coordinator, and the pool.

The balance-sheet treatment. At the member firm, hosting fees are an operating expense in the period, and the subscription is an ordinary membership cost — no capitalisation, no intangible, no argument with the auditor. At the consortium, the apprentice contract is a fixed-term employment obligation with a matching receivable from hosting members; where the consortium pre-funds a cohort, the unrecovered balance sits as a deferred cost recovered over the remaining contract, which is a schedule an auditor will recognise because it is the same one used for any prepaid service contract. The one thing not to attempt is capitalising a person. You capitalise the contract, never the skill, and the distinction survives audit precisely because it is the honest one.

The counterparty. The consortium, and behind it the joint subscription base. For a first facility, the natural anchor is a sector association or a large firm already at scale in training, which contributes the coordinator and takes the first year's administrative cost in exchange for first rotation choice.

The first ninety days.

Day	Action	Artifact
1–15	Compute your own payback month: cost per trainee-year, productive contribution by year, cumulative crossing	The payback curve, one page
16–30	Measure retention honestly — proportion still present twelve months after qualification, three cohorts back	The tenure number
31–45	Compare P to T. If P < T, train on your own book and stop here	The decision memo
46–60	If P > T, recruit three to five members whose skill families overlap and whose rotations differ	Heads of terms
61–75	Set the subscription: the aggregate shortfall divided across member headcount	The subscription schedule
76–90	Incorporate; sign the first cohort; publish the payback month to the apprentices themselves	Signed contracts

The number that decides it. One inequality, and it goes on the front page:

    P  =  months until cumulative contribution exceeds cumulative cost
    T  =  expected months of retained tenure, measured not assumed

              Train alone when   P < T
              Pool when          P > T

Switzerland: P is 27.7 and the contract runs thirty-six months, so P clears T by more than eight months and retention is simply not a variable. Germany: P is thirty-six months of training plus sixty of recovery — ninety-six — against a 53 per cent chance of still having the person at month forty-eight. The gap is held closed by institutions, not by any single firm's arithmetic, which is exactly why a single firm outside those institutions should not attempt it alone.

And the sensitivity that matters. Re-run P with the depreciation rate of the skill you are actually building. If δ is near ten per cent a year, no design with a payback beyond about seven years is worth modelling, whatever the retention assumption — the asset is half gone. Shorten the payback or pool. The depreciation rate is not a footnote to this decision. It is the decision.

APPRECIATIVE QUESTIONS

Twelve, for a room

Discovery — what is already working

Think of someone here who learned most of what they know on this site. What did we do, specifically, that made that possible — and who else knows we did it?
Where in our work does someone become genuinely productive far earlier than the job description assumes? What is different about how that role is taught?
Which of our people have left and taken something valuable with them — and what did we get from them before they went that we never counted?

Dream — what becomes possible

If every person starting here were told their payback month on day one, what would change about the first year — for them, and for the person teaching them?
Imagine our skills register as a real asset schedule, with a depreciation rate beside each family. Which line would surprise the board most?
If we were known as the best place in this sector to be trained and then leave, what would that do to who applies to us?

Design — what we build

Which of our training designs could have its payback moved six months earlier without reducing what is learned — and what exactly would have to change in week one?
Which three firms nearby have skill needs that overlap ours enough to share a cohort, and differ enough that rotations would teach more than we can alone?
What would we have to measure, starting this month, for P and T to be numbers rather than opinions a year from now?

Destiny — how it holds

What is the first sign we would see if our training had quietly stopped being worth it, and who in this building would see it first?
What would it take for the person we train at forty-five to get the same quality of formation as the person we train at eighteen?
If the people who built this system all left, what would still be true in three years — and what would we want to be true?

WORKS CITED

Acemoglu, D. and Pischke, J.-S. (1998). "Why Do Firms Train? Theory and Evidence." Quarterly Journal of Economics, 113(1), 79–119.

Acemoglu, D. and Pischke, J.-S. (1999). "Beyond Becker: Training in Imperfect Labour Markets." Economic Journal, 109(453), F112–F142.

Acemoglu, D. and Pischke, J.-S. (1999). "The Structure of Wages and Investment in General Training." Journal of Political Economy, 107(3), 539–572.

Arrazola, M. and de Hevia, J. (2004). "More on the Estimation of the Human Capital Depreciation Rate." Applied Economics Letters, 11(3), 145–148.

Becker, G. S. (1964). Human Capital: A Theoretical and Empirical Analysis, with Special Reference to Education. National Bureau of Economic Research.

Beicht, U., Walden, G. and Herget, H. (2004). Kosten und Nutzen der betrieblichen Berufsausbildung in Deutschland. Bundesinstitut für Berufsbildung.

Cavendish Square Holding BV v Talal El Makdessi [2015] UKSC 67.

Deming, D. J. and Noray, K. (2020). "Earnings Dynamics, Changing Job Skills, and STEM Careers." Quarterly Journal of Economics, 135(4), 1965–2005.

Germany. Berufsbildungsgesetz (2005), §12 — Nichtige Vereinbarungen.

Hanushek, E. A., Schwerdt, G., Woessmann, L. and Zhang, L. (2017). "General Education, Vocational Education, and Labor-Market Outcomes over the Life-Cycle." Journal of Human Resources, 52(1), 48–87.

Mincer, J. (1974). Schooling, Experience, and Earnings. National Bureau of Economic Research / Columbia University Press.

Muehlemann, S. and Wolter, S. C. (2014). "Return on Investment of Apprenticeship Systems for Enterprises: Evidence from Cost-Benefit Analyses." IZA Journal of Labor Policy, 3:25.

Patrignani, P., Conlon, G., Dickerson, A. and McIntosh, S. (2021). The Impact of the Apprenticeship Levy on Apprenticeships and Other Training Outcomes. Centre for Vocational Education Research, Discussion Paper 034.

Psacharopoulos, G. and Patrinos, H. A. (2018). "Returns to Investment in Education: A Decennial Review of the Global Literature." Education Economics, 26(5), 445–458.

Schönfeld, G., Wenzelmann, F., Dionisius, R., Pfeifer, H. and Walden, G. (2010). Kosten und Nutzen der dualen Ausbildung aus Sicht der Betriebe. Bundesinstitut für Berufsbildung.

Schweri, J., Mühlemann, S., Pescio, Y., Walther, B., Wolter, S. C. and Zürcher, L. (2003). Kosten und Nutzen der Lehrlingsausbildung aus der Sicht Schweizer Betriebe. Rüegger Verlag.

Strupler, M. and Wolter, S. C. (2012). Die duale Lehre — eine Erfolgsgeschichte, auch für die Betriebe. Ergebnisse der dritten Kosten-Nutzen-Erhebung der Lehrlingsausbildung aus der Sicht der Betriebe. Rüegger Verlag.

Wolter, S. C. and Ryan, P. (2011). "Apprenticeship." In Hanushek, E. A., Machin, S. and Woessmann, L. (eds), Handbook of the Economics of Education, Volume 3. Elsevier, 521–576.

Arbejdsgivernes Uddannelsesbidrag (AUB), Denmark. Employers' Reimbursement Fund, established 1977; contribution rates as published by the fund.

Note on figures. Every number above is computed in lib/verify/V_05.py and reproducible with python3 lib/verify.py V.05, which prints its inputs, their units and their sources before it prints a result. The Swiss payback is computed from the fourth cost-benefit survey's aggregate costs and output, distributed across the three years using the productive-value shares published by Strupler and Wolter, and is reported under two cost profiles rather than one. The depreciation rate implied by Deming and Noray is computed on the premium, not on one plus the premium; the module prints both and shows why the second is wrong.