§1 · The papers
Peter F. Colwell, Money, Inflation, and a Macro Model, version 2.0, 5 July 2022 (48 pages; I read the abstract, introduction, the opening of the model and the conclusions). It argues that inflation is an increasing, convex function of the rate of money growth and builds a simple monetarist steady-state model around that. A. W. Phillips, The Relation between Unemployment and the Rate of Change of Money Wage Rates in the United Kingdom, 1861–1957, Economica, November 1958 (I read the passage giving the fitted equation). Joseph E. Gagnon and Christopher G. Collins, Low Inflation Bends the Phillips Curve, Peterson Institute working paper 19-6, April 2019 (abstract and introduction only).
§2 · The identity
Colwell starts from the equation of exchange, M·V = P·Q, which is true by the definition of velocity (V = PQ/M). In rates of change it reads m + v ≈ p + q, where m, v, p, q are the growth rates of money, velocity, prices and real output. In discrete steps the exact form is (1 + m)(1 + v) = (1 + p)(1 + q).
An invented example: money grows 6%, velocity 1%, prices 3%. Then (1.06)(1.01) = 1.0706, so nominal activity grows 7.06%, and real output grows 1.0706 / 1.03 − 1 = 3.94%. The additive rule gives 6 + 1 − 3 = 4.00%. The 0.06-point gap is the product term, small at low rates and growing as rates rise.
§3 · The shape of Colwell’s curve, on invented numbers
Colwell hypothesises that for positive money growth, inflation p(m) rises at an increasing rate, with a positive intercept p(0). I do not use his estimates. To see what such a shape implies, take an invented form p(m) = a + b·m² with a = 0.02 and b = 2.5, and, as in his steady state, constant velocity (v = 0), so that real growth is q = m − p(m).
| Target | Rule | Money growth m | Inflation p | Growth q |
|---|---|---|---|---|
| Best growth-to-inflation ratio | m = √(a/b) | 0.0894 | 0.0400 | 0.0494 |
| Best growth minus inflation | m = 1/(4b) | 0.1000 | 0.0450 | 0.0550 |
| Best growth | m = 1/(2b) | 0.2000 | 0.1200 | 0.0800 |
The ordering is the point: the ratio-maximising money growth is lowest, the net-benefit one is in the middle, and the growth-maximising one is highest, and the growth-maximising point comes with a much higher inflation rate (12%) than the other two. The ordering holds whenever a·b < 1/16, which these numbers satisfy (a·b = 0.05). Colwell reports the same qualitative ordering for his estimated function, with Friedman’s 4% rule nearest the ratio maximum and the Fed’s 2% target nearest the growth maximum; I could not check those correspondences, which depend on his data.
§4 · Phillips’s curve, as printed
For 1861–1913 Phillips fitted y + 0.900 = 9.638 x−1.394, where y is the percentage rate of change of wage rates and x is the percentage unemployment; equivalently log(y + 0.900) = 0.984 − 1.394 log x. The two forms agree: log10 9.638 = 0.984.
| Unemployment x (%) | 2 | 4 | 5.48 | 6 | 8 | 10 |
|---|---|---|---|---|---|---|
| Wage change y (% per year) | 2.77 | 0.50 | 0.00 | −0.11 | −0.37 | −0.51 |
Wage change is zero at an unemployment rate of about 5.48% (the fitted curve crosses zero there), and the curve never falls below −0.9% per year however high unemployment goes: that is the shift constant. The curve is convex and flattens: the fall in y between x = 2 and x = 3 is 1.58 points, and between x = 8 and x = 9 it is only 0.08. These are properties of the fitted equation, not claims that wages behave this way today.
§5 · Putting them side by side
Both curves are convex, and both papers handle rates that can be negative with a shifted logarithm: Phillips uses log(y + 0.900), and Colwell’s estimation tables use ln(p + 0.025). Beyond that they differ. Colwell’s curve has inflation rising with money growth; Phillips’s has wage change falling with unemployment. They relate different variables, in different periods and countries, with different data.
Shared functional form, not a shared invariant. Nothing here derives either curve from the other or shows that one constant governs both. The resemblance is in shape and in the use of a shift; it is not evidence that they are two views of one law.
Gagnon and Collins argue that the Phillips curve may be nonlinear when inflation is low, so that the economy could have spent about twenty years on a flat part of the curve; their abstract says the data are consistent with that and with a fall in the curve’s slope, and that the next few years might decide between the two. That paper is from 2019. What the data have shown since is not checked here.
§6 · What is not established
- Colwell’s empirical results: not reproduced and not read in full. I read the abstract, introduction, the opening of the model and the conclusions, and I did not read the estimation sections line by line. His figures for his rules are his claims.
- The status of the Colwell paper: it appears to be an author-circulated working paper; I saw no journal citation. Treat its claims as unreviewed.
- The numbers in §3 are invented to show a shape, and the form a + b·m² is my choice, not his.
- Phillips’s data and fit: only the printed equation was used, which is the fit in his section on 1861–1913; his later periods (1913–1948 and 1948–1957) were not used. Later work, including the expectations critique, is not covered.
- Gagnon and Collins: abstract and introduction only; their estimates were not read.
- Anything about current inflation, unemployment or monetary policy.
Sources
Three PDFs held by the author, read 2026-11-05: Colwell (2022), Phillips (Economica, November 1958), Gagnon and Collins (PIIE 19-6, 2019). Checked by wp139-verify.py: every number in §2 to §4.