Haute Lumière
Commerce · III.08 · MMXXVI · daylight
TEN CONCEPT BRIEFS · Chapter III.08 — Velocity, Circulation, and Health
One page each. A reader who reads only these ten pages has the chapter.
BRIEF 1 — The Quantity Equation
The idea. Money spent equals money received. Written across a whole economy for a whole year, that trivial fact becomes the most reused equation in monetary economics.
Fisher (1911) M · V_T = P · T every transaction
income form M · V = P · Y only final output — nominal GDP
Cambridge form M = k · P · Y, k = 1/VThe income form is the one everybody uses, and it already contains a choice: P · Y is nominal GDP, which counts only final output. Intermediate trade, second-hand goods, house purchases and every financial transaction are outside it. So the V in the income form is not the number of times a dollar was spent. It is the number of times a dollar was spent on things the national accounts count.
Worked example. In 2025Q2, nominal GDP was 30,331.0 billion dollars and M2 was 21,900.0 billion. V = 30,331.0 / 21,900.0 = 1.3850.
Why it matters. The equation is an identity — true by construction, like a balance sheet balancing. It becomes a theory only when you add assumptions: that V is stable, that M is set independently, that Y does not respond to M. Each of those is a separate empirical claim, and none of them is supplied by the equation itself.
You already know this because you have seen a revenue figure that was technically correct and told you nothing, once you learned what had been excluded from it.
BRIEF 2 — Velocity Is a Residual
The idea. P, Y and M are each produced by a measurement procedure. V is produced by none. It is defined as nominal GDP divided by M, which means there is exactly one degree of freedom in the equation and V is it.
Nobody has ever timed a dollar. There is no survey of dollar turnover, no sampling frame, no instrument. The velocity series is a division.
Worked example. M2 velocity fell from 2.192 in 1997Q3 to 1.100 in 2020Q2, a fall of 49.8 percent. Take logs and the fall decomposes exactly: ln(19,913.1/8,542.2) = 0.8464 for nominal income against ln(18,109.0/3,897.0) = 1.5362 for the money stock. The difference is −0.6898, and e^−0.6898 = 0.5017. Nominal income multiplied by 2.331; M2 multiplied by 4.647. That pair of numbers is the halving. Nothing about speed was observed.
Why it matters. Any sentence of the form circulation is slowing is the sentence the money stock grew faster than nominal income, translated into a vocabulary that implies a diagnosis. The translation smuggles in the diagnosis.
You already know this because you have met a ratio in a management pack that moved entirely because its denominator was redefined, and watched a room discuss it as though something had happened.
BRIEF 3 — The Cambridge k, or the Holding Period
The idea. Invert velocity and the same arithmetic stops implying anything. k = 1/V is the share of a year's income held as money — the Cambridge cash balance. It is a statement about choice, not speed.
Worked example.
1997Q3 V = 2.192 k = 0.4562 of a year = 5.5 months of income held
2019Q4 V = 1.426 k = 0.7014 = 8.4 months
2020Q2 V = 1.100 k = 0.9094 = 10.9 months
2025Q2 V = 1.385 k = 0.7220 = 8.7 monthsWhy it matters. Velocity halved and people and firms chose to hold twice as much liquidity relative to income are the identical measurement. The first sounds like a circulatory disease; the second sounds like a decision with reasons — precautionary saving, low opportunity cost of holding cash, deposit insurance, a pandemic. Only the second phrasing invites you to ask what the reasons were, and the reasons are the only actionable part.
You already know this because you keep more cash in your current account when you are uncertain, and you would be surprised to hear that described as your personal economy slowing down.
BRIEF 4 — Which M? The Aggregate Problem
The idea. V has a denominator, and the denominator is chosen. Change the choice and every velocity figure changes, with no transaction occurring.
Worked example — the cleanest natural experiment in monetary statistics. On 24 April 2020 the Federal Reserve amended Regulation D, removing the six-per-month transfer limit on savings deposits; from the next H.6 release savings deposits were reported inside M1.
2020Q1 M1 4,002.0 $bn V = 21,538.0 / 4,002.0 = 5.382
2020Q2 M1 16,206.0 $bn V = 19,913.1 / 16,206.0 = 1.229A fall of 77.2 percent in one quarter. Estimate old-basis M1 for 2020Q2 at 5,000.0 billion and the split is: real factor 0.7400, definitional factor 0.3085, so 79.6 percent of the log fall is the definition. Walk the estimate from 4,600.0 to 5,400.0 and the definitional share only moves between 85.3 and 74.4 percent.
Why it matters. William Barnett's long argument that simple-sum aggregates add imperfect substitutes at equal weight — and that a Divisia index should be used instead — is not a technicality. It is the observation that the denominator of every velocity figure is an accounting convention.
You already know this because you have seen a like-for-like sales figure change the moment somebody changed what counted as a store.
BRIEF 5 — The Retention Rate
The idea. Forget speed. Ask instead: of a pound received by a local counterparty, what proportion is respent with another local counterparty? That proportion — call it r — is observable, because it is a ratio of actual payments to actual named parties.
Worked example. Civic Economics' Andersonville study of Chicago retailers found independent retailers respending about 0.48 of revenue inside the local economy against about 0.14 for chain retailers. Preston's anchor institutions moved from £38.0 million of £750.0 million landing in the city (5.07 percent) to £111.0 million of £616.0 million (18.02 percent) — £73.0 million a year more, out of a base that had fallen by £134.0 million.
Why it matters. A retention rate requires a boundary, and the boundary is half the number. A postcode district, a travel-to-work area, a named list of parishes — write it down, publish it, and write the rule for a counterparty with several addresses, because that is where the measurement actually breaks. A figure whose boundary can move is a figure that can be produced to order.
You already know this because you already know that "local" means something different to the council, the chamber of commerce and your own supplier list, and that nobody has ever written down which one is meant.
BRIEF 6 — Leakage Is a Geometric Series
The idea. A pound lands. A proportion r is respent locally; a proportion r of that is respent again. The local income generated per pound is the sum of a geometric series, and the series has a closed form.
after n rounds L(n) = (1 − rⁿ)/(1 − r) = 1 + r + … + r^(n−1)
in the limit L = 1/(1 − r)
LM3 is L(3) LM3 = 1 + r + r²Worked example.
r = 0.48 LM3 = 1 + 0.48 + 0.2304 = 1.7104 L = 1.9231
r = 0.14 LM3 = 1 + 0.14 + 0.0196 = 1.1596 L = 1.1628A pound generates £1.71 of local income at the higher rate against £1.16 at the lower — a gap of 55.1 pence over three rounds, and a ratio of 1.4750. In the limit the ratio is 1.6538.
Why it matters. This is arithmetic you can do in your head, and it replaces an argument that is usually conducted in adjectives. Applied to Preston's extra £73.0 million, rounds two and three are worth £51.86 million at r = 0.48 and £11.65 million at r = 0.14 — a difference of £40.21 million a year, produced by routing and not by quantity.
You already know this because you understand compound interest, and this is compound interest with the sign of the exponent flipped.
BRIEF 7 — LM3, and What It Leaves Out
The idea. LM3 stops at three rounds because that is where survey response rates collapse, not because the economy stops. The share of the true multiplier that three rounds capture is L(3)/L = 1 − r³.
Worked example.
r = 0.14 LM3 = 1.1596 L = 1.1628 captured 99.7 %
r = 0.48 LM3 = 1.7104 L = 1.9231 captured 88.9 %
r = 0.80 LM3 = 2.4400 L = 5.0000 captured 48.8 %The truncation bites hardest exactly where retention is highest. The instrument the local-economy literature relies on systematically understates its own case.
And the warning in the same arithmetic. Back a uniform r out of NEF's published local-food LM3 of 2.50 — solve 1 + r + r² = 2.50 — and you get r = 0.8229, which would be 82.3 percent retained at every round, far above any single-round rate anybody has measured. Field LM3 does not assume a uniform r; it measures a different proportion at each round. The two methods are not interchangeable, and presenting a model as a measurement is what discredits this field.
You already know this because you have seen a three-year payback quoted for an asset with a fifteen-year life, and understood that the number was conservative and misleading at the same time.
BRIEF 8 — The Convexity of Retention
The idea. The value of one more point of retention is not constant. It is dL/dr = 1/(1 − r)², which rises steeply as r approaches one.
Worked example.
r = 0.22 marginal value 1.644
r = 0.45 marginal value 3.306
r = 0.80 marginal value 25.000A firm with £12,000,000 of annual third-party spend at r = 0.22 generates £15,220,800 of local income over three rounds (LM3 = 1.2684). At r = 0.45 it generates £19,830,000 (LM3 = 1.6525) — a difference of £4,609,200 a year. Round two alone moves from £2,640,000 to £5,400,000.
Why it matters. Procurement targets are almost always written as flat percentage-point increases — five points a year — which treats every point as equal. The arithmetic says the last points are worth the most, so a flat target will stop precisely where the value was about to begin.
You already know this because you know the difference between a system that is ninety-five percent reliable and one that almost never fails, and that the last few points cost and deliver more than all the others combined.
BRIEF 9 — The Circulation–Efficiency Threshold
The idea. A multiplier counts transactions. Transactions are the cost of getting something done, not the getting-it-done. If a higher-retention supply chain uses proportionally more real input p to deliver the same basket, then circulation per unit delivered is LM3/(1 + p), and there is a p at which the higher multiplier stops being worth having.
Worked example.
1.7104 / (1 + p) = 1.1596 → p* = 0.4750 = 47.5 %
penalty p circulation per unit delivered verdict
none 1.7104 above the chain
10.0 % 1.5549 above
20.0 % 1.4253 above
47.5 % 1.1596 breakeven
60.0 % 1.0690 belowWhy it matters. This is the honest negative of the constructive half of the chapter, and it is also reassuring: at a ten percent price premium the local route still delivers 1.5549 against 1.1596, so in practice you are nowhere near the threshold. But a community wealth programme that cannot state its own breakeven penalty is not yet a programme — above p*, the extra circulation is the sound of the same basket costing more to produce, and the town is poorer with a better multiplier.
You already know this because you know that a business with more invoices is not thereby a better business.
BRIEF 10 — Cardiac Output, and the Category Error
The idea. The claim that a faster-circulating currency is healthier is imported from physiology. Check it in physiology's own units and it inverts.
cardiac output = heart rate × stroke volume
rest 60 beats/min × 70 ml = 4,200 ml/min
tachy 100 beats/min × 42 ml = 4,200 ml/minIdentical delivered flow. Rate up 66.7 percent, stroke volume down 40.0 percent. A clinician reading that pair calls it compensated shock, not fitness. The health variable is output, not rate.
The economic translation. The analogue of cardiac output is real goods and services delivered; the analogue of heart rate is velocity. Over the window in which M2 velocity fell 49.8 percent, US real output per person rose from 43,551 to 62,434 chained 2017 dollars — 43.4 percent. Faster money at constant delivery is the tachycardia, not the athlete.
Where the intuition was right. Inside one firm a pound really does carry a timestamp, and the measurement is exact: a supplier billing £2,000,000 a year on 60-day terms holds a £328,767 receivable; at 10 days it is £54,795. But notice what the benefit actually is. It is the £273,973 of released working capital — a stock, not the move from 6.08 to 36.50 turns a year, which is the artefact.
You already know this because you have met someone who was very busy and not, on inspection, getting more done.