Where Regeneration Costs More

Volume IV — Production and Regeneration

THE PLATE

A woman in a cream cardigan reading an open book on her lap, soft light from the side. — Plate IV.11The Invoice, Read Twice.A number that will not come down is still a number you can build on. The mistake is not paying it. The mistake is calling it temporary.

THE LETTER

This volume has spent ten chapters showing you where making things differently pays. It has been honest about the timing and it has shown its working, and the arithmetic has held: waste is a revenue line, loops recover value, soil is a capital asset, and a great many regenerative moves are cheaper than the incumbent once the accounting boundary is drawn properly.

This chapter is about the cases where that is not true and will not become true.

There are four of them in this volume's territory, and each is well measured. The organic yield gap narrows and then stops. Recycled content in some polymer streams costs more at every volume anyone can foresee. Repair loses ground against manufacture every single year and the gap widens. And on the question that decides how much of the planet stays wild, the published evidence favours farming a smaller area harder rather than a larger area gently.

You should know these before somebody hostile tells you, because the person who already knows is in a completely different position from the person who is hearing it. And there is a second reason, which is the actual subject of the chapter: a premium that will not fall is a different kind of object from a premium that will, and it wants a different instrument. Almost every failure of regenerative finance we have watched has been an instrument built for a temporary gap, wrapped around a permanent one, and then quietly abandoned when the gap did not close on schedule.

So: the arithmetic first, without softening. Then who should pay, on what principle, with which instrument. Then the rule a firm uses on a Monday when it is looking at a cost that is never going to come down — computed, so you can run it on your own numbers before you finish this page.

— The Editors

DISCOVERY

What is already working

Start where somebody is already paying a permanent premium well, because it is being done, and the people doing it have solved things you would otherwise have to invent.

Copenhagen's public kitchens. The city feeds around seventy thousand people a day — schools, nurseries, care homes, staff canteens — and since the early 2000s it has moved that food to roughly ninety per cent organic. The interesting part is not the share. It is that the change was made inside the existing food budget, through Københavns Madhus, the city's own food house, by a redesign of what was cooked rather than a redesign of what was bought: less meat, far less waste, more seasonal vegetables, more cooking from raw ingredients and less purchase of prepared food. The organic premium was real and permanent. It was paid for out of a different line.

That is the first working pattern, and it is the one most often missed: a permanent premium on an input can be absorbed by a change in the recipe. The firm that only asks "can we afford organic flour" has not yet asked the question that Copenhagen answered.

Norway's bottle loop. Norway charges an environmental tax on beverage packaging that falls as the national return rate rises, and disappears entirely once return exceeds ninety-five per cent. Infinitum, the industry-owned deposit company, runs the system; return rates for PET bottles sit in the low nineties. Read the mechanism carefully, because it is the cleanest instrument in this whole chapter. The state did not subsidise recycling. It taxed the substitute — the container that does not come back — and then handed the industry a way to extinguish the tax by collective performance. The premium is paid by the person who declines to return the bottle, and the deposit gives them a reason not to decline.

The UK offshore wind auctions. From 2015 the United Kingdom paid for offshore wind through contracts for difference awarded by competitive auction: a strike price, a two-way settlement against a market reference, fifteen years. It was designed to pay a premium that everyone serious believed was structural. We will do the arithmetic on what happened next in the next movement, because it is the most important number in the chapter.

Aluminium. And then the control case, which teaches you to tell the two situations apart. Primary aluminium is made by electrolysis at roughly fourteen kilowatt-hours per kilogram. Remelting scrap takes about 0.7 kWh/kg — a factor of twenty, a saving of 95 per cent of the energy. There is no premium here and there never was one. Nobody had to mandate an aluminium loop, write a standard for it or run an auction: the loop is simply the cheap way to get the metal, and it closed itself.

Four cases, and the pattern that matters is the contrast between the last one and the first three. Where the loop is cheaper, it closes without anybody being clever. Where it is not, every single closed loop you can point at is being held closed by an instrument. A procurement standard. A tax on the substitute. An auction. A recipe change. Not by conviction, and not by communication.

So the first practical move, before any arithmetic, is a sorting exercise, and it takes an afternoon. Take every regenerative substitution your firm is considering and put each into one of three boxes.

Already cheaper, and the market has not noticed. Rare but real, usually because the incumbent cost is hidden in an overhead. Just do it.
More expensive now, cheaper later, at a volume you can name. This is a financing problem. Volume IV's earlier chapters are about this box, and the shared-savings facility from I.01 is the instrument.
More expensive now and at every volume you can foresee. This chapter. Different instrument, different owner, different conversation, and — this is the part nobody expects — potentially the most defensible position of the three.

The single most expensive error in regenerative finance is putting a box-three item in a box-two instrument. It buys you three years of patience and then a reputation for having been wrong, which costs more than the premium ever did.

THE ARITHMETIC

Four premiums that do not close, and one that did

The organic yield gap. Three meta-analyses, three samples, three methods. Seufert, Ramankutty and Foley (2012) put organic yields 25 per cent below conventional across 316 comparisons. De Ponti, Rijk and van Ittersum (2012) put them at 80 per cent of conventional — a 20 per cent gap — across 362. Ponisio and colleagues (2015), on the largest sample at 1,071 comparisons, found 19.2 per cent. The unweighted mean of the three is 21.4 per cent and they disagree by 5.8 percentage points, which is the honest width of the answer rather than a reason to prefer one.

Carry the largest sample forward. A 19.2 per cent gap means land per tonne of 1 / (1 − 0.192) = 1.2376, so 23.8 per cent more land for the same food. That is the sentence to hold, because the yield gap is a land premium before it is ever a price premium, and the land premium cannot be paid by anybody's willingness to pay. A shopper can pay more. A shopper cannot pay hectares.

Does it close? Partly, and then it stops. In the same study, organic systems using multi-cropping closed the gap to 8.7 per cent and those using extended crop rotations to 9.0 per cent. That is 54.7 per cent of the gap removed by better management, and 45.3 per cent of it still standing under the best practice anybody has published.

The price side is more comfortable than expected and should worry you anyway. Crowder and Reganold (2015), across 55 crops in 14 countries over 40 years, found the break-even premium — the price uplift at which organic matches conventional profitability — at 5 to 7 per cent, against actual observed premiums of 29 to 32 per cent. The premium is over-paying the gap by a factor of 4.14× to 6.4×. Read that as a market statement, not a moral one: the entire margin of organic agriculture sits in a price premium set by scarcity of supply, and premiums set by scarcity of supply do what all such premiums do.

Recycled content, where the loop does not pay. Since 1950, humanity has made around 8,300 Mt of virgin plastic resin and generated 6,300 Mt of plastic waste. Nine per cent of that waste has ever been recycled — 567 Mt — twelve per cent incinerated, seventy-nine per cent accumulated (Geyer, Jambeck and Law, 2017). Food-grade recycled PET flake in Europe has repeatedly traded above virgin bottle-grade resin. Take indicative figures of £1,100/t virgin against £1,450/t recycled: a spread of £350/t, or 31.8 per cent.

Now ask whether volume closes it. Put a generous learning rate of 12 per cent per doubling on the recycled route against 3 per cent on virgin resin. The gap multiplies by 0.88 / 0.97 = 0.9072 per doubling, so crossover arrives after 2.84 doublings — about 7.1 times today's volume. That is reachable, and for a clean single stream like deposit-return PET it is the right answer: this is a box-two problem.

Then look at the term inside it that has no learning rate at all. Collection is a vehicle and a driver arriving at a door. At an assumed £1.10 per stop and 0.45 kg of target polymer per kerbside household visit, kerbside collection costs £2,444/t. The same £1.10 at a return point yielding 2.60 kg costs £423/t — 5.78 times cheaper. And the kerbside collection term on its own is 6.98 times the entire spread the learning curve was asked to close.

That is the anatomy of a permanent premium: a cost term whose driver is not cumulative experience. Manufacturing costs fall with experience. Collection costs fall with density, and density is set by where people live and how they shop. No amount of industry maturity moves it. This is why deposit-return works and kerbside mixed-plastic collection does not, and it is a structural fact rather than a failure of effort.

Repair against a falling manufacture cost. A device at £200 with an £80 out-of-warranty repair starts at a repair-to-new ratio of 0.40. Manufacture falls at perhaps 4 per cent a year in real terms; repair is labour and rises with real wages at perhaps 1.5 per cent. The ratio multiplies by 1.015 / 0.96 = 1.0573 every year — a divergence of 5.73 per cent per year — and reaches 0.50 in 4.01 years and parity, where repairing costs as much as replacing, in 16.45 years. There is no crossover in this model and there cannot be one: the two series diverge monotonically, forever. Baumol wrote it down in 1966 about orchestras, and where the manufactured substitute is a lithium-ion product falling at 13 per cent a year (Ziegler and Trancik, 2021), the divergence is far steeper than the illustration.

What survives it is a price band rather than a sector. The same divergence applied to an £1,800 device with a £200 repair starts at a ratio of 0.111 and takes 27.0 years to reach 0.50. The repair economy is not a movement that failed at the cheap end. It is an economics that only ever worked above a price line, and the line rises every year.

Sparing against sharing. Take a 1,000 km² landscape with 500 km² farmed at conventional yield. The same food from a system with the 19.2 per cent gap needs 618.8 km², leaving 381.2 km² of habitat against 500 km² — the gap costs 118.8 km², which is 23.8 per cent of the habitat. The marginal price is 5 km² per percentage point of yield gap at a gap of zero, rising to 7.66 km² per point at a gap of 19.2 per cent. The cost per point rises as the gap widens.

Against that, organic farmland genuinely carries more life: Tuck and colleagues (2014) found 34 per cent higher species richness on organic farms, Bengtsson and colleagues (2005) about 30 per cent. In this landscape the shared system carries 65.8 per cent more farmland-generalist capacity, while sparing carries 31.2 per cent more habitat. Apply a species-area exponent of z = 0.25 and specialist persistence is 0.841 under sparing against 0.786 under sharing — a 6.56 per cent specialist loss from choosing to share.

Both lines are true simultaneously and the debate stops being a debate once you say which guild you are counting. Sharing carries more common species. Sparing carries more of the species that occur in no farmland at any intensity — which is the set the word conservation was coined for, and which is what Phalan and colleagues (2011) found empirically in Ghana and northern India. Kremen (2015) answers that real landscapes are not a binary and that a spared reserve surrounded by sterile monoculture is not what sparing's advocates model. She is right, and the arithmetic above is still the arithmetic.

Now the honest positive, because one of these was wrong. In 2012 the UK's own Offshore Wind Cost Reduction Task Force set £100/MWh by 2020 as a stretch target, and the industry regarded the premium above that as structural. The first contracts-for-difference auction in 2015 cleared offshore wind at £114.39/MWh in 2012 money. The second, in 2017, cleared at £57.50 — 42.5 per cent below the 2020 stretch target, hit 3 years early, and 37.8 per cent below the £92.50/MWh strike that Hinkley Point C had been granted the year before. By 2019 it was £39.65, a real decline of 23.3 per cent a year across four years, and by 2022 £37.35, a total fall of 67.3 per cent.

The instrument built to pay a permanent premium is the instrument that ended it, and the reason is worth carrying out of this chapter: the premium was set by an auction rather than by an estimate, so it had a clock on it. Everybody in 2012 was looking at a volume problem wearing the costume of a permanent one. Which means the first duty of anyone who writes the word permanent is to say what would falsify it — and for offshore wind, in 2017, the answer arrived.

And here is the cut. Work out what a premium with no crossover is actually worth. A crossover confers an advantage — say £100 per unit per year — for about 6 years, until imitators cross the same line and competition removes it. At 9 per cent that is an annuity factor of 4.4859 and a present value of £448.59 per unit. A permanent margin of the same present value is only £40.37 per unit per year: 40.4 per cent of the temporary one.

So a durable margin worth two-fifths of a temporary one is worth exactly the same money — and a cost that will not come down cannot be waited out by an entrant either. Every chapter in this volume that promises a crossover is describing an advantage that arbitrage removes by year six. This chapter, which promises none, is the one describing a position that holds. The permanent premium is not the weak case in the book. Priced correctly, it is the moat.

DREAM

What becomes ordinary

In the economy that has absorbed this, the word premium has stopped being an accusation and become a line item with an owner.

Every material a firm buys carries two numbers rather than one: its price, and the premium over its incumbent substitute, with that premium classified as temporary or permanent and the classification dated and signed. Temporary premiums sit with the treasury, financed against the crossover year like any other bridge. Permanent premiums sit somewhere else entirely — with marketing if a price test has demonstrated customers will pay it, with compliance if a levy avoids it, with the board if the honest answer is that the firm has decided to buy something it values and is booking the cost as what it is. Nobody is confused about which, and nobody is embarrassed about the third case, because a declared donation is respectable and a disguised one is not.

Public buyers name the premium they will carry and hold it steady long enough for somebody to build a factory against it. A recycler can read a city's procurement standard and know what tonnage is contracted for seven years, which is what turns an intention into a lender's credit memo. The standard is written as an outcome rather than as a technology, so the supplier who finds a cheaper route keeps the saving, and the standard does not have to be rewritten when they do.

Levies fall on substitutes rather than on virtue. The bottle that does not come back carries the cost; the returned one does not. Because the charge is extinguishable by performance, the industry treats the system as an asset rather than a fine, and runs it well.

Repair has stopped being a moral campaign and become a well-drawn commercial segment. Everyone knows where the price line is, manufacturers above it design for repair because their customers will pay for a twelve-year product, and below the line the argument has moved — sensibly — from repairability to material recovery and to the design decisions that make recovery cheap. Nobody is asking a volunteer to spend an hour of skilled labour restoring an item worth less than the hour.

And the yield-gap question is asked landscape by landscape rather than in general. Which fields are farmed hard and which land is left alone is a decision somebody makes deliberately, in a room, with the species lists in front of them and the arithmetic on the table — rather than an outcome that falls out of a thousand separate purchasing decisions nobody ever added up.

Most of all: the sentence this will get cheaper is now a claim that carries a date and a falsifier, the way any other forecast does. When it is right, the instrument retires itself. When it is wrong, somebody notices in year two rather than year nine.

DESIGN

Who pays, and the four instruments that spread it

The allocation principle, first, because the instruments follow from it.

A permanent premium should be paid by whoever holds the avoided cost.

That is the whole rule and it resolves almost every argument about fairness by turning it into a question of fact. If the avoided cost is a public one — habitat, emissions, a landfill that a council pays for — the public holds it and the public should pay, through a standard or a levy. If the avoided cost is private and lands on the buyer as a differentiated product they will genuinely pay more for, the buyer holds it and the price carries it. If the avoided cost accrues to the incumbent substitute's producer in the form of a cost they are not bearing, put the charge there. And if nobody holds an avoided cost at all, then the premium is buying something the firm values for its own reasons, which is allowed and is booked as a choice rather than smuggled in as an investment.

Instrument one — the procurement standard. A public or anchor buyer commits to a minimum specification over a stated tonnage and a stated term. Three design rules make the difference between a standard that builds capacity and one that produces a price spike. Write it as an outcome, not a technology, so the supplier who finds a cheaper route keeps the margin. Give it a term long enough to finance against — five to seven years, because nobody builds a sorting line for a twelve-month contract. And phase it against real supply, because a mandate that arrives before capacity does not create capacity; it bids up the price of the existing supply and hands the windfall to incumbents.

Instrument two — the contract for difference. A strike price, a published reference index, two-way settlement, a fixed term. The producer is insulated from the spread and the buyer's cost is known. Two features are what make it the best instrument in this chapter. It is two-way, so when the market moves in the producer's favour the money comes back — which is what makes it a hedge rather than a subsidy, and what makes it survivable politically. And where the strike is set by competitive auction rather than by an estimate, the instrument discovers its own obsolescence, which is exactly what happened to offshore wind between 2015 and 2019.

Instrument three — the tariff. A border adjustment charges imports the carbon cost that domestic producers bear, so a domestic standard does not simply export the production it was meant to clean. The EU's Carbon Border Adjustment Mechanism entered its transitional reporting phase on 1 October 2023 and its definitive regime from 1 January 2026. The design point is that a tariff is the instrument you need when the premium is real and your competitors are outside your jurisdiction — without it, a domestic standard is a relocation policy.

Instrument four — a levy on the substitute. The most elegant of the four, and the most often mis-sized. The UK's Plastic Packaging Tax charges £217.85 per tonne (2024/25 rate) on packaging containing less than 30 per cent recycled plastic. Meeting the threshold at a £350/t premium costs £105/t and avoids £217.85/t with certainty: a net gain of £112.85/t, and every rational producer goes straight to thirty per cent.

Now the number that designers of such levies rarely compute. The maximum premium this instrument can carry is £217.85 / 0.30 = £726.17/t, and it carries nothing at all above the threshold, because above thirty per cent the tax is already avoided and the instrument has stopped paying. A levy with a threshold buys exactly the threshold. That is not a flaw — it is a precise, cheap and predictable purchase of a specific quantity of behaviour. But a government that wants fifty per cent recycled content and writes a thirty per cent threshold will get thirty per cent, and will be surprised.

The sequence. Standard first, to create demand that can be financed. Contract for difference second, to de-risk the supply that answers it. Levy third, to make the substitute carry its own cost once an alternative exists — because a levy imposed before supply exists is a tax with no exit and it will be repealed. Tariff last and only where trade exposure is material, since it is the slowest to build and the most contested.

DESTINY

How it holds when nobody is pushing

Three things keep a permanent premium paid, and they are not the three that keep a temporary one paid.

It is classified, in writing, and the classification is reviewed. A premium recorded as permanent, reviewed annually, next review March survives a change of chief executive. A premium carried in somebody's conviction does not. The review is the load-bearing half: classifications are claims, and a claim nobody re-examines becomes furniture.

Its payer is named and the payer holds the avoided cost. A premium paid by whoever happens to have budget is paid until that budget is cut, which is usually the second bad quarter. A premium paid out of a line that would otherwise go on a levy, a landfill charge or a customer discount is paid for as long as the arithmetic holds, by somebody with a self-interested reason to keep paying it.

It has a stated falsifier. Write down, when you classify a premium as permanent, the observation that would change your mind: the volume at which the spread would close, the wage-to-manufacture ratio that would make repair pay, the yield-gap figure that would flip the sparing arithmetic. Then check annually. Offshore wind is the case that earns this paragraph — three years after everyone serious called the premium structural, an auction proved otherwise, and the firms that had written down a falsifier moved first.

And here is where this fails. It fails when a firm classifies a premium as permanent because that is the comfortable answer — permanent means nobody has to try — and stops looking for the process change that would close it. It fails when the premium is genuinely permanent and the firm pays it anyway at full substitution, above p*, out of a budget that cannot carry it, and the programme is cancelled entirely in the next downturn rather than reduced to the tonnage that pays. It fails when a levy's threshold is treated as an ambition rather than as what it is — a purchase order for exactly that threshold. And it fails, most quietly, when nobody wrote the falsifier down, the world changed, and the firm went on paying a premium that had stopped existing because the review had become a formality.

DELIGHT

What it feels like

There is a specific relief in the meeting where somebody finally says the number out loud and nothing bad happens. The premium is £350 a tonne. It is not going away. Now — what do we want to do about that?

The room changes. The conversation had been circling for two years around a hope nobody would state plainly, and hope is exhausting to maintain in a meeting. The moment the figure is on the table the exhaustion lifts, because a known cost is an object you can arrange things around and a hoped-for saving is not.

And there is a second, better pleasure that arrives later. It is the pleasure of being the only firm in your category that knows its own number. Everybody else in the sector is still waiting for the cost to come down, which means they are not building anything against it, which means that when the procurement standard lands or the levy arrives they will be bidding for the same scarce supply you contracted four years ago at a price you understood.

Honesty about a permanent cost turns out to be an operational advantage rather than a confession, and it feels like the difference between carrying something and putting it down.

OPERATIONALIZE THIS

At the level of finance

The rule, first, because everything else is sizing.

Pay a permanent premium p per unit if and only if:

      p  <  p*  =  m  +  s  +  τ · λ / (λ + w)

  m  realised revenue premium, from an actual price test — never a survey
  s  avoided direct cost: disposal, energy, compliance, insurance
  τ  the levy you will pay per unit if you do not switch
  λ  hazard rate of that levy arriving, per year
  w  your cost of capital

The term λ / (λ + w) is the regulatory arrival discount — the fraction of a future levy you may credit today. It is the only honest way to put an expected regulation on a page that a CFO will sign, and it has the right properties: it goes to one as the levy becomes imminent and to zero as it recedes.

Worked, on the polymer case. m = £60/t from a genuine price test. s = £40/t in avoided disposal and compliance. τ = £217.85/t, the current levy. λ = 0.15, an expected arrival in 6.67 years. w = 0.09. Then the discount is 0.625, the credit is £136.16/t, and:

   p*  =  60  +  40  +  136.16  =  £236.16 / t
   p   =  £350 / t
   p / p*  =  1.48x

p > p*. Full substitution is therefore a donation of £113.84 per tonne, and the correct action is not to abandon the programme but to buy exactly the tonnage that pays: the thirty per cent threshold, where £105/t of premium extinguishes £217.85/t of certain levy. Book the difference honestly. A donation is permitted in this house. A donation described as an investment is not.

The instrument: a bilateral contract for difference on the material spread.

Structure. Buyer and recycler contract a volume — say 12,000 t/yr — against a strike spread in £/t over a published virgin index. Monthly two-way settlement against the observed spread. Term 7 years, matched to the recycler's debt tenor, because the contract exists to make that debt bankable.
Settlement. At a strike of £320/t against today's observed £350/t, settlement is −£360,000/yr: the recycler pays the buyer. At £240/t the buyer pays the recycler £960,000/yr. The two-way leg is what makes this a hedge rather than a subsidy, and it is what lets a treasurer approve it.
Exposure. Worst case for the buyer is the spread going to zero: £320 × 12,000 = £3.84m/yr, £19.33m discounted at 9 per cent over the term. That figure goes on the front page. A contract whose worst case is not stated is not a contract, it is a hope with signatures.
Balance-sheet treatment. A physically settled offtake for the buyer's own manufacturing consumption is generally eligible for the own-use exemption under IFRS 9, and stays off the derivative books. The same economics written as a financially settled spread contract is a derivative, marked to market through profit and loss. The accounting is decided by how you write it, not by what it does — agree the treatment with your auditors before drafting, not after.
Counterparty. The recycler directly where you can take the credit; a bank as intermediary where you cannot. Where a public buyer is in the chain, the strongest version is a tripartite standard-plus-CfD: the public body sets the outcome standard, the firm contracts the spread, the recycler builds.
The number that decides it. Sign only where the strike spread is at or below p. At a strike of £320/t against a p of £236.16/t the ratio is 1.355× and the answer is no at full volume — so the contract is written on the threshold tonnage the levy pays for, and the balance of the book stays on virgin until p* moves, which it does the moment τ rises or λ shortens.

The first ninety days.

Day	Action	Artifact
1–15	Sort every substitution into the three boxes	The classification, signed and dated
16–30	For each box-three item, compute `p*` on your own numbers	The `p*` sheet
31–45	Run a real price test for `m` — a live price, a real basket	The measured `m`, not a survey
46–60	Write the falsifier for every "permanent" classification	The falsifier register
61–75	Draft the spread CfD on threshold tonnage; agree own-use treatment	Term sheet plus audit memo
76–90	Take the donation line to the board as a donation	One page, one number, one owner

APPRECIATIVE QUESTIONS

Twelve, for a room

Discovery — what is already working

Where are we already paying a premium on purpose, and paying it well — who decided, and what did they see that the rest of us did not?
Which of our loops closed without anybody having to be clever about it, and what made that one different from the loops we are still arguing about?
When have we changed a recipe rather than a supplier to absorb a cost — and what else could that move work on?

Dream — what becomes possible

If every material we buy carried its premium and its classification beside its price, which three decisions would get easier immediately?
Imagine we are the only firm in our sector that knows its own permanent numbers. What does that let us contract for that our competitors cannot?
If a declared donation were entirely respectable here, what would we choose to buy, and what would we stop pretending was an investment?

Design — what we build

Who in this business holds the avoided cost for each premium we pay, and are they the one paying it?
What outcome standard could we write that a supplier could finance against — and how long would it have to run to be worth their while?
Where is our threshold set, and is the thing we want the same as the thing we are actually buying?

Destiny — how it holds

What would have to be observed for us to change our mind about a premium we have called permanent, and who checks it?
Which of our "temporary" premiums has been temporary for more than five years, and what would we do differently if we reclassified it today?
If the person who believes in this most left tomorrow, which of these commitments would still be in the reporting pack in a year?

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Note on figures. Every number in this chapter is computed in lib/verify/IV_11.py and printed there with its inputs, its units and its source. Figures marked indicative or assumed — market prices for PET, collection costs per stop, learning rates, the levy hazard rate — are inputs a reader is invited to replace; every conclusion is stated as a function of them, so a better input gives a better answer from the same lines.