What Is Money? A Promise Written Twice

Your banking app says $40.

Where is it? Not in a vault with your name on it. Your app shows one line: you're owed $40. The bank's books hold the other line: it owes you $40. The $40 is those two lines agreeing.

That isn't a figure of speech. The Bank of England's introduction to money says "Money today is a type of IOU".¹ An IOU is a promise, and a promise sits in two places: the books of the one who is owed and the books of the one who owes.

Take that picture seriously and some strange facts about money stop being strange. Holding it costs nothing. Issuing it earns the issuer nothing by itself. Interest is only a transfer between two people. And the books fix no price. Those are the first half of the question. The second half, where value comes from, comes out of the same two lines.

The short answer

Money is a promise written twice. In the holder's books it's something they're owed. In the issuer's books it's something the issuer owes: two records of one claim. (The issuer is whoever owes it: your bank, or the central bank behind a banknote.)

While the two records agree, the ledger charges nothing for holding the money, at any amount and for any length of time. Issuing it earns the issuer nothing by itself. Interest is only a transfer between the two ends of a claim, and the books fix neither which way it runs nor how fast. The books fix no price. The two sides' figures bound a range of prices both accept, and a point inside the range is a rule people add.

Value isn't a substance stored in goods or coins. Reality keeps a permanent record of every event, because the basic laws of physics never destroy information. We call it the ledger. It holds comparisons between people and never amounts, so there's nowhere in it to store a worth. An amount matters in two ways: it decides which acts are possible, and it fills the reserves that pay for repair. Production adds value by turning inputs into something people can use, and every unit of material is still counted.

Here's the whole answer in seven steps. The rest of this post takes them one at a time, with the reason for each and a small example with the numbers worked through.

1. Money is a promise written twice. One record in the holder's books, one in the issuer's.
2. Holding it costs nothing. The cost sits in a gap between two records, never in the size of a holding.
3. Issuing it earns nothing, and interest is only a transfer. The books fix neither its direction nor its rate.
4. A price is a range. Both sides' figures bound the prices they both accept. A point inside is a rule people add.
5. Value isn't stored in things. The ledger reads ratios. Production adds what you can do, never material.
6. A bigger total never settles it for a member. A total can rise while a person loses.
7. The economy and the ethics are one ledger. So an act that moves a person's side takes that person's write.

Why this matters: "What's the right price?" and "what is it worth?" sound like questions with a number for an answer. The ledger's answers are shapes: a range for a price, and no line at all for a worth. And once money is booked as a promise, the ledger fixes what it costs to hold, who can earn from it and who has to agree to change it. Each of these comes with a small example, and every number is on the page.

Three facts this post stands on

Each comes with its reason. A Favor Isn't a Yes works through the second one at more length, and Part 1 covers the first one and the smallest event.

First, nothing that happens is ever erased. Every event leaves a complete trace, so the full record of what happened is part of reality now. The past still exists. The reason starts in known physics: the basic laws never destroy information. In quantum mechanics this is called unitarity. The present fixes the past exactly, and running the laws backward recovers every earlier state. Our own work adds a second, separate reason: the record only grows. Each new entry is added with every earlier entry kept, and writing a new entry never alters an earlier one.²

Second, every relation has two ends, and each end keeps its own books. The smallest thing that can happen is one thing being told apart from another, a single yes or no. You're doing it right now: every perception and every measurement tells one thing from another. Telling two things apart takes information about both, so an entry is a comparison, and a comparison needs both of its terms and reads nothing else. That makes the ledger's unit a relation between two ends, read as one number: the ratio of its two terms. Each end keeps its own record of the relation and writes only in its own books. An act may move a relation only when both of its ends write on it. If an act moved it on one end's write alone, the other end's record of it would be false, and nobody else can write in that end's books to fix it. And since two true records of one relation agree, a gap between two records means at least one is false.³

Third, a comparison has a cost, and one formula gives it. Call the ratio of a relation's two terms x, with x = 1 at balance. The cost of the comparison is

J(x) = (x + 1/x) / 2 - 1

The cost has to be zero only when the two terms match, grow as they differ, and come out the same whichever term you put first. Those requirements, one rule for how the costs of combined comparisons add, and one calibration (how sharply the cost rises near a match) leave exactly this formula.⁴ Try it. At x = 1: (1 + 1)/2 - 1 = 0. At x = 2: (2 + 1/2)/2 - 1 = 1/4. At x = 1/2: (1/2 + 2)/2 - 1 = 1/4 again. J measures how far a relation is from balance. It isn't a score of anyone's wealth or happiness.

Step 1: Money is a promise written twice

The claim. In the books, money is a claim on an issuer. One end of the relation is owed and the other end owes. The holder's record says "I'm owed this." The issuer's record says "I owe this." They're two records of one relation.

How we know. This is how money is booked, and the Bank of England describes it the same way: each type of money "features on the balance sheets of at least two different groups, because each is an asset of one group and a liability of another."¹ Your bank balance is an asset to you and a liability to the bank. A banknote is the same kind of thing, with the central bank as its issuer.

This post starts from that booking. It's a definition, set down as the form of money, and the consequences in Steps 2 and 3 are proved on it.⁵ Each record is drawn as a small book. The books are a drawing. The records are real: each end of a relation holds its own.

Watch a payment. You have $40 at the bank, and a shop has $10. You pay the shop $15. Nothing is carried anywhere. The bank moves its promise: you and the bank both write $15 less owed to you, and the shop and the bank both write $15 more owed to the shop.

When the move is done, every pair of records still agrees: your 25 against the bank's 25, the shop's 25 against the bank's 25. The Bank of England describes a card payment the same way: the banking sector reduces the amount it owes the payer and increases the amount it owes the shop.¹

Why does the shop accept the bank's promise at all? The Bank of England's answer is trust: "everyone in the economy trusts that it will be accepted by other people in exchange for goods and services."¹ That's a question about people. This post is about what the promise is and what follows once it's booked.

Next, and why: what does it cost to hold a promise like that? One number answers it, and the answer is the first strange fact.

Step 2: Holding it costs nothing

The claim. While the two records agree, holding the money costs nothing, at any amount and for any length of time. The cost lives in a gap between records, never in standing still.

How we know. The relation between you and the bank compares two records, and its cost is J of their ratio. Your record says 40 and the bank's says 40. The ratio is 1, and J(1) = 0.

Change the amount: 5 against 5, or 5 million against 5 million. What the ledger reads from a relation is the ratio of its two ends' figures, never the figures themselves.³ The ratio is 1 either way, so the cost is 0 either way.

Change the time. At the next step in the writing of the record the two records still agree, so the ratio is still 1 and the cost is still 0. Add up a day of zeros, a year of them, a thousand years, and the total is 0.

Break the agreement. Say your record says 40 and the bank's says 20. The ratio is 2, and J(2) = 1/4. The comparison costs 1/4 as long as the gap stands, and a gap means at least one of the two records is false. Zero cost doesn't prove the records are true, though: two matching records can both be wrong. Which is true is read from what happened between the two ends. Sorry Doesn't Rewind the Tape shows how.

This one is plain arithmetic: J(1) = 0. It still tells you where the cost lives: in a gap between two records, not in the size of a holding.

Next, and why: if holding costs nothing, can the one who issues the promise at least earn from issuing it?

Step 3: Issuing it earns nothing, and interest is only a transfer

The claim. Issuing money can't make the issuer richer. And interest is only a transfer between the two ends of a claim: one end's net position rises by exactly what the other's falls, nobody else's changes, nobody can book it alone, and the books fix neither its sign nor its size.

Why issuing earns nothing. Every party's books balance, because each entry has a counter entry on the same party's books. (That's double-entry bookkeeping, the rule the books are kept by here.) A party's net position is what it's owed minus what it owes. Now take an issuer whose books hold nothing but claims: the promises it has made and the claims it holds. In the drawing below, the bank owes 25 to you and 25 to the shop, and Ana owes the bank 50. Its books balance among those claims alone, so its net position is exactly 0. It stays 0 through every consistent history of moves, payments included.⁵

An issuer can't get ahead by writing more promises. To end up ahead it needs a line of its own that isn't a claim on anyone, something it owns outright, where a gain can be booked as its own. Then it's a party like any other. The Bank of England's paper has the same shape. If everyone pooled all their assets and debts together, the financial ones, money included, "would cancel out, leaving only the non-financial assets."¹

What interest is. Bob owes Alice $100, and y of interest is booked on it. There are two ways to book it. In the first, the claim grows by y on both books: Alice books y as income and Bob books y as expense, and no cash moves. In the second, Bob pays Alice y in cash through the bank, and the claim stays at $100. These are different books: in one Alice holds a bigger claim, in the other she holds more cash. But add up each person's net position either way and the answer is the same. Alice's rises by y. Bob's falls by y. Nobody else's moves, the bank's included.

No one can book interest alone. Booking it changes a claim, and an act may move a relation only when both of its ends write on it: one end changing its line for a claim alone would leave the claim's two records disagreeing. An entry one party writes alone can only move value between lines it owns outright, and that changes nobody's net position.

And the books don't say what y is. Both bookings are consistent for y = 6, for y = 600, for y = 0 and for y = -6, and for a negative y the transfer runs the other way. The cost J is the same whichever record is larger, so it doesn't say which way a transfer should run either. The books fix neither the sign of interest nor its rate. They carry whatever the two ends write. They don't treat interest as a charge for standing still, because standing costs nothing (Step 2). In the books, interest is a transfer.

Next, and why: the strangest thing missing so far is a price. Where does the price of a thing come from?

Step 4: A price is a range

The claim. The books fix no price. The two sides' figures bound a range of prices that both sides accept, and nothing in the books picks a point inside it. A point needs a sharing rule, and people add that rule.

How we know. A sale has two sides, and each side's book carries a figure for the deal. Say you're selling a bike. Your book carries $100. The buyer's carries $400. They're the two ends' figures for one deal, and they disagree: the ratio is 4, and the gap costs J(4) = 9/8.

To close the gap, both books move to one price, p. Moving a book from its own figure to p is a concession, and a concession has a cost too: J of p divided by that side's own figure. The books don't hand us a rule for which prices count as acceptable, so we state one: a side accepts p when its concession costs no more than the disagreement it ends.

Try p = $150. Your concession costs J(150/100) = J(1.5) = 1/12, about 0.083. The buyer's costs J(150/400) = J(3/8) = 25/48, about 0.52. Both are under 9/8, so both accept. The algebra gives the whole answer: the prices both sides accept are exactly the ones from $100 to $400.⁶ At $100, the buyer's concession costs exactly what the disagreement does. At $400, yours does. Past either end, one side says no.

Which price in the range? The books don't say. A point needs a rule for sharing, and rules differ. "Both sides concede equally," meaning the same concession cost on each side, lands where J(p/100) = J(p/400). That's p × p = 100 × 400, so p = $200, the geometric mean of the two figures. "Split the difference" lands on $250. Each is a rule people choose. The books choose neither, and the geometric mean is a statistic of the two figures, not a price the ledger sets.

A stricter rule narrows the range. Say a side accepts p only when its concession costs no more than half the disagreement it ends. Then the prices both sides accept run from about $145 to $276. Under either rule the answer is a range, not a point. So haggling isn't the books failing. It's the part they leave to you.

The shape is old. In 1881 F. Y. Edgeworth argued that two traders contracting only with each other have a whole stretch of settlements to choose from, and that which one they reach is "arbitrary in the absence of arbitration."⁷ The ledger gets the same shape from two books.

Next, and why: if the books fix no price, what do they say a thing is worth?

Step 5: Value isn't stored in things

The claim. Value isn't a substance stored in goods or coins. The ledger reads a ratio for each relation and never an amount, so a balanced holding costs nothing at any size (Step 2). An amount matters in two ways: it decides which acts are possible, and it fills the reserves that pay for repair.³ Production adds value by turning inputs into forms people can use, and every unit of material is still counted.

How we know. Take a small economy where every step is counted. Alice holds one raw unit and one unit of effort. Bob holds one ready meal. The raw unit is uncooked food, enough for two meals. Bob can eat now. Alice can't: her food is right there, but no meal exists until someone spends effort to make one.

Alice produces. The raw unit and the unit of effort become two ready meals. That's the whole technology of this example: one raw unit plus one effort makes two ready units, and units can be stored, retrieved, handed over and eaten.⁸

Count the material, with a raw unit counting as two because it holds two meals' worth. Before: 2 + 1 = 3. After: Alice's 2 ready meals plus Bob's 1 = 3. Alice's effort is spent, and the effort total, available plus spent, is still 1. In this example every step, in every order, keeps both totals. Nothing was created. What changed is what Alice can do. She can eat now, and she couldn't before.

What a stock can do depends on what's around it. A stock with one raw unit and no effort looks empty, because all it can do is wait. Hand it one unit of effort and it can produce. Hand the same effort to a truly empty stock and it can't. In this example, two stocks allow exactly the same actions in every situation exactly when four quantities agree: raw, ready, stored and effort. An idle stock isn't worthless just because it can't act alone. (Four is this example's number, not a law about economies.)

So what did Alice gain? Not material: the count stayed at 3. She gained a use. That's how production adds value. It turns inputs into forms people can use, and every unit of material is counted while it does.

Next, and why: an amount decides what can be done. A total of amounts can mislead, and the next step shows how.

Step 6: A bigger total never settles it for a member

The claim. A total can rise while one person loses. Two plans can show identical totals while one breaks a member's reserve. So no total settles it for a member.

How we know. Two small cases, each exact.

A total up, a member down. Go back to Alice and Bob in Step 5. From the same start, Alice produces and then takes Bob's meal. Every step is physically possible. The ready meals rise from 1 to 3, and Bob's fall from 1 to 0. Material and effort are conserved the whole way, and the meals available still went up while Bob lost his. Nothing in the steps says Alice is allowed to take the meal. Being physically possible isn't being allowed.⁹

The same totals, a different member. Here the amounts that matter are reserves, what each person draws on to keep going. In a second small economy each person keeps a reserve, and the rule is that every reserve stays above zero at every step. Each step costs each person 2 from their reserve. Eating one unit of food adds 1 to the eater's reserve, and one unit of work costs the worker 1. (These rates are this example's own, chosen to keep the numbers small.) There are three piles of material: one to cook from, one Alice eats from and one Bob eats from. They start at 2, 3 and 1, and the reserves start at 6 for Alice and 5 for Bob.

Two plans begin the same way. Alice does one unit of work and turns 2 units of the first pile into 2 more in her own pile. That leaves the piles at (0, 5, 1) and the reserves at (3, 3). Then the plans split.

Both plans end with the same food left, the same total eaten (4) and the same total in reserve (2). Every total agrees. Plan A keeps both reserves above zero. Plan B drops Bob's to zero. No rule that reads only those totals can tell the two plans apart, so none can say whether every reserve stayed above zero. Keep each person's own reserve in the report and the plans separate at once.¹⁰

It's the small, exact version of a familiar question: if the economy grew, why am I worse off? In these constructed economies a total can rise while a member falls, and no report of totals can tell.

Next, and why: if totals can't settle it, what does? The ledger that reads the economy also reads the ethics, and that's the last step.

Step 7: The economy and the ethics are one ledger

The claim. An economic change and an ethical action are one change of the same relations. So no act may move one person's side without that person writing it, and the ledger has no second tally of value beside the cost.

How we know. A person's burden is the cost of the relations they're an end of. Each end carries a relation's full cost, and the ledger's own total counts each relation once.

Now take what actually reaches a person against what they need. That's a comparison with a ratio: actual divided by required, balanced at 1. Say Bob needs 2 meals and 1 reaches him. The ratio is 1/2, and the cost is J(1/2) = 1/4. Alice hands him a meal. Now 2 reach him against the 2 he needs: ratio 1, cost 0. Both ends carry the relation's cost, so both burdens fall from 1/4 to 0, Alice's too, although she gave up a meal.

An act changes a relation by multiplying its ratio by a factor. Here the factor is 2, which takes 1/2 to 1. That isn't special to this example. Take any finite set of relations and any comparison you supply, a positive actual and a positive required for each. Applying to each relation the factor after divided by before gives exactly the ledger after, and no other factor does. So the economic change and the ethical act are one change of one ledger. And the unit cancels: count meals or calories and each ratio, and so each factor, comes out the same.¹¹

Now the consequence. Three people stand in a chain: Alice, Bob and Carol. The relation between Alice and Bob sits at 4, and the one between Bob and Carol at 1. J(4) = (4 + 1/4)/2 - 1 = 9/8 and J(1) = 0, so the whole chain costs 9/8.

Move both relations to 2. The relation between Alice and Bob is relieved from 4 to 2, and the one between Bob and Carol is strained from 1 to 2. The product of the two ratios stays 4, since 4 × 1 = 2 × 2, so the strain is spread out, not removed. And spreading it costs less. Each relation now costs J(2) = 1/4, and the whole chain costs 1/2. The whole saves 5/8.

Look at Carol. Her only relation was balanced, with cost 0. Now it costs 1/4. Her burden rose from 0 to 1/4.

Who has to agree? An act may move a relation only when both of its ends write on it, and A Favor Isn't a Yes teaches that rule with its reason.¹² The relation between Bob and Carol has Carol at one end. No act may move it without her write. So this move, which saves the whole 5/8, needs Carol. Her write isn't a cost that the saving can outbid. Any finite charge for moving someone's side without their write is outweighed by some such move, because the relief such a move can buy grows without bound. So consent comes before cost: the rules for acts compare costs only among acts that both ends of every moved relation write, and no saving to the whole licenses an act that moves one member's side without that member writing it. Right and Wrong Aren't Opinions lays out the rules for acts.

Then where is the value? Can the ledger carry a second number, the value of a relationship or of knowing, added to the cost? No. A comparison reports one thing: J of its ratio. Add a continuous value term that isn't a multiple of J, and the total is no longer what any comparison reports.¹³ So the ledger has one cost and no price tags: no number on a thing that says what it's worth. Where the ledger does count an amount, it counts it for what the amount makes possible and for the reserve it fills: the acts it covers, and the repairs its reserve pays for. That's where worth shows up in the ledger, and it isn't a number stored in the thing.

Everything above about economic changes is proved for any comparison you supply. The step that ties it to the physical ledger isn't derived yet: which physical comparison an economic need is. That is, that each economic comparison, a need's actual against its required or a claim's two figures, is a relation on the ledger with the ledger's multiplier as its ratio, and that a relation's cost is exactly what its comparison reports, with no value term added.

The World Underneath the Labels outlines the same result in its section on economics.

What's still open

• Which physical comparison an economic need is (Step 7). For any comparison you supply, an economic change is the ethical action on the same ledger. Still to derive: that each economic comparison, a need's actual against its required or a claim's two figures, is a relation on the ledger with the ledger's multiplier as its ratio, and that a relation's cost is exactly what its comparison reports, with no value term added.
• What's taken as given. Money's form, the issuer's promise booked twice (Step 1). The rule for which prices a side accepts (Step 4). The small economies of Steps 5 and 6, built so every number can be checked. None of these is a claim about a particular real economy.

Two things look like gaps and aren't: the point inside the price range (Step 4) and the sign and rate of interest (Step 3). The books don't contain them, and that is a result. A rule people choose supplies them.

The whole trip, in one breath

Money is a promise written twice: one record in the holder's books and one in the issuer's, two records of one claim. While they agree, the ledger charges nothing for holding it, at any size and for any time, because the cost of a relation sits in a gap and a ratio of 1 has no gap. The issuer earns nothing by issuing, because its books balance among claims and only a line of its own that isn't a claim can hold a gain. Interest is only a transfer between the two ends of a claim. The two figures of a deal bound the prices both sides accept, and a point inside is a rule people add. Value isn't stored in things. The ledger reads ratios, an amount decides what can be done and what a reserve can pay for, and production turns material into use without creating any. A bigger total never settles it for a member, because an economic change is an ethical action on the same relations, so an act that moves a person's side takes that person's write. That's how reality keeps its books: one ledger, two ends to every relation, and no price tags.