Public Cash MoneyThis chapter does not prescribe P.C.M. It corrects the standard Equation of Exchange to include variables the mainstream framework omits, demonstrates that the corrected formula describes historical reality more accurately than the standard one, and introduces the Return Coefficient R as a formal instrument for distinguishing productive from non-productive public expenditure.
Davide Serra · Systems Analyst · publiccashmoney.com · May 2026
Published as Open Source. Peer review actively welcomed.
Prerequisite: Chapter 1 of this framework. Mathematical level: secondary school algebra.
Abstract
The standard Equation of Exchange, MV = PQ, formalized by Irving Fisher in 1911 and adopted as a foundational identity by monetarist and New Keynesian frameworks, omits two variables that are structurally significant in any debt-based monetary system: tax obligations (T) and interest on outstanding debt (I). This chapter demonstrates that the omission is not trivial: the standard formula systematically underpredicts debt growth, while the corrected formula MV = PQ + T + I produces projections that correspond more closely to documented historical trajectories. We further extend the corrected formula to include a government expenditure parameter G×S, and introduce the Return Coefficient R as a formal instrument for classifying public expenditure by its net economic contribution. We conclude with an empirical application of the corrected formula to the post-Bretton Woods trajectory of US national debt.
1. The Standard Formula and Its Predictions: the Equation of Exchange, as formalized by Irving Fisher in “The Purchasing Power of Money” (1911), states:
M × V = P × Q
The standard Equation of Exchange. M = money supply, V = velocity of circulation, P = price level, Q = real output. P×Q = nominal GDP.
In its monetarist interpretation, developed principally by Milton Friedman, this identity is combined with two assumptions: that V is approximately stable in the short run, and that Q is determined by real factors independent of monetary conditions. Under these assumptions, changes in M translate directly into proportional changes in P — the foundational claim of the quantity theory of money. The formula is presented as a complete description of the monetary economy. Its implicit prediction regarding debt is straightforward: if nominal GDP (P×Q) grows, and if fiscal revenues grow approximately proportionally with nominal GDP, then debt sustainability is a function of the relationship between the interest rate r and the growth rate g. The condition r < g, formalized by Blanchard (2019), is presented as the condition under which debt remains sustainable without fiscal adjustment. This prediction has a testable empirical implication: in a period of sustained nominal GDP growth, debt-to-GDP ratios should stabilize or decline, provided r < g holds.
2. The Empirical Failure of the Standard Prediction
The post-Bretton Woods trajectory of US national debt constitutes a natural experiment of unusual duration and completeness. From 1944 to 2026, the United States operated as the issuer of the global reserve currency under a debt-based monetary system. The data are comprehensive, publicly available, and uncontested.
Empirical record — US economy 1944 to 2026
US nominal GDP grew from approximately $2 trillion (1944) to $27 trillion (2026): a 13-fold increase over 82 years, representing sustained nominal growth averaging approximately 3.2% annually. US national debt grew from approximately $260 billion (1944) to $39 trillion (2026): a 150-fold increase over the same period, representing average annual growth of approximately 6.3%. The ratio of debt growth to GDP growth: approximately 11.5x. Debt grew at approximately twice the rate of nominal GDP throughout the post-Bretton Woods period, including extended periods during which the r < g condition was satisfied.
Sources: US Treasury Fiscal Data; Federal Reserve FRED database; Congressional Budget Office historical budget data.
The standard formula offers no structural explanation for this divergence. If MV = PQ is a complete description of the monetary economy, and if nominal GDP grew at 3.2% annually, debt growth at 6.3% annually represents a persistent anomaly that the standard framework cannot account for without reference to discretionary political decisions — excessive spending, insufficient taxation, poor fiscal management. This chapter argues that the divergence is not anomalous. It is structurally predicted by the corrected formula, and the standard formula’s failure to predict it is a direct consequence of its omission of T and I.
3. The Corrected Formula: MV = PQ + T + I
The standard formula omits two variables that are structurally present in any debt-based monetary system operating under a fiscal regime.
Variable T — Tax obligations generated by monetary circulation: In any jurisdiction with an income or value-added tax, each transaction in which the monetary unit changes hands in exchange for goods or services generates a tax obligation. This obligation is proportional to the nominal value of the transaction (P×Q) and to the prevailing tax rate τ.
T = τ × (P × Q) = τ × M × V
Critically: T is a function of V. As the velocity of circulation increases — as the same monetary unit changes hands more frequently — the tax obligation generated per unit of money in circulation increases proportionally. T must be paid in money. It cannot be settled in real goods. This creates a structural demand for monetary units that is absent from the standard formula.
Variable I — Interest obligations on outstanding debt: In a debt-based monetary system, every unit of money currently in circulation was created through a lending operation carrying an interest obligation. As demonstrated in Chapter 1, the total interest obligation I on outstanding debt D at prevailing rate r is:
I = r × D where D > M always (from Chapter 1, Theorem 1)
I grows continuously as long as D > 0 and r > 0. It is independent of V and independent of current economic output. It represents a structural drain on the monetary system that exists regardless of the level of economic activity.
The corrected identity therefore reads:
M × V = P × Q + T + I
The corrected Equation of Exchange. T = tax obligations generated by circulation. I = interest obligations on outstanding debt. Both T and I must be satisfied from M×V. Neither was included in the standard formula. The corrected formula has an immediate and significant implication: the net monetary surplus available for real economic activity (P×Q) is not M×V, but M×V minus the structural drains T and I. Increases in V do not simply increase nominal output proportionally — they simultaneously increase T proportionally and leave I unchanged. The net effect on available purchasing power is always less than the gross effect of the velocity increase. The standard formula says: increase V and nominal output increases proportionally but the corrected formula says: increase V and nominal output increases, but the tax drain T increases proportionally, while I remains, compounding on the growing debt base. The net available for real transactions is always less than M×V and this gap is structural, not discretionary: it exists whether or not the government spends prudently and It exists whether or not fiscal policy is conservative or expansionary so it is the consequence of the architecture, not the behavior of the actors within it.
4. Extension: The Government Expenditure Parameter G×S
The corrected formula MV = PQ + T + I captures the structural drains inherent in a debt-based monetary system under a fiscal regime. To complete the description of fiscal dynamics, we introduce a government expenditure parameter.
Parameter G×S — Government expenditure
Let S denote total government expenditure in a given period, and G ∈ [0,1] denote the government’s propensity to spend relative to its maximum feasible expenditure. In a debt-based system, expenditure beyond tax revenues T is financed through new debt issuance, which increases D and therefore increases I in subsequent periods.
Additional debt generated per period: ΔD = G×S − T (when G×S > T) and the extended corrected formula becomes:
M × V = P × Q + T + I + G × S
This extension permits a formal statement of three important propositions:
Proposition 1 — Debt growth is never zero in a debt-based system
Even with G = 0 (zero discretionary spending beyond what tax revenues cover), I remains positive and growing as long as D > 0. Debt cannot reach zero from within the system without an external monetary injection. This follows directly from Chapter 1, Theorem 1.
Proposition 2 — Higher G accelerates debt growth
As G increases toward 1, the additional debt generated per period (G×S − T) increases, adding to D and therefore to I in subsequent periods. Higher spending propensity produces faster debt accumulation, not proportionally but compoundingly — because each increment of new debt generates additional interest that compounds on the growing base.
Proposition 3 — The r < g condition is necessary but not sufficient for sustainability
The Blanchard condition r < g is formulated within the standard MV = PQ framework. In the corrected framework, even with r < g, the structural drain T + I grows with V and with D respectively. Long-run sustainability requires not only r < g but also that the growth of T + I does not exceed the growth of P×Q. Historically, this condition has not been maintained: as demonstrated in Section 2, debt grew at twice the rate of nominal GDP throughout the post-Bretton Woods period, even during extended periods of r < g.
5. The Return Coefficient R: A Formal Instrument for Expenditure Classification
The most consequential analytical gap in both the standard and corrected formulas is the treatment of government expenditure as a homogeneous quantity. In both MV = PQ and MV = PQ + T + I + G×S, public spending appears as a single variable S without regard to the economic character of the expenditure it represents.
This chapter proposes the Return Coefficient R as a formal instrument for correcting this gap.
Definition — The Return Coefficient R
For any category of public expenditure with cost C, the Return Coefficient R is defined as the ratio of the total economic value generated by the expenditure over its economically relevant time horizon to the immediate cost of the expenditure:
R = V_generated / C where V_generated = net present value of economic output attributable to the expenditure
Three categories of expenditure are formally distinguishable:
R > 1: Productive expenditure. Generates more economic value than its cost.
R = 0: Neutral or destructive expenditure. Generates no economic return.
R < 0: Value-destroying expenditure. Destroys existing economic capacity.
5.1 Empirical Calibration of R
R cannot be calculated with precision for any specific expenditure category in advance. The future economic value generated by any investment is subject to uncertainty, context-dependency, and measurement challenges. The purpose of R in this framework is not precise calculation but formal classification: to distinguish, at the level of economic logic, expenditure that creates productive capacity from expenditure that does not.
Historical evidence provides reasonable order-of-magnitude estimates for several categories:
| Expenditure category | R estimate | Evidence base |
|---|---|---|
| University education (G.I. Bill, 1944) | R ≈ 6.9 | Congressional Joint Economic Committee analysis (1988): $6.90 returned per $1 invested, via increased output and tax revenues over 35 years. Source: Subcommittee on Education and Health, Joint Economic Committee, December 1988. |
| Transport infrastructure (roads, railways) | R > 1 | Extensive infrastructure economics literature documents positive returns through logistics cost reduction, market expansion, and productivity gains. Precise R varies by project and context. |
| Public health infrastructure | R > 1 | WHO and World Bank estimates consistently document positive economic returns from healthcare investment through workforce productivity and reduced chronic disease burden. |
| Military hardware (deployed in conflict) | R = 0 | Deployed military hardware generates no economic output. Its value is destroyed at the moment of use. No productive capacity is created. |
| Active warfare | R < 0 | War destroys existing capital — infrastructure, housing, productive capacity, human capital — that required prior investment to create. Net economic contribution is negative. This finding is robust across historical studies of conflict economics. |
5.2 The Structural Indifference of Debt-Based Financing
The critical observation enabled by the Return Coefficient R is the following: in a debt-based monetary system, the financing cost of public expenditure is independent of R. A government that issues debt to fund a university (R ≈ 6.9) and a government that issues debt to fund a military campaign (R < 0) pay the same interest rate on the same debt instrument. The $1.x design bug documented in Chapter 1 does not distinguish between expenditure categories. The interest obligation I accumulates at the same rate regardless of whether the expenditure generated economic value sufficient to service that obligation.
Formally: for expenditure with cost C financed by debt at rate r, the total obligation at time t is C×(1+r)^t, independent of R.
This structural indifference has a compounding consequence over time. Expenditure with R > 1 generates economic value V_generated > C that could, in principle, service the debt obligation C×(1+r)^t if V_generated grows faster than (1+r)^t. Expenditure with R = 0 generates no value to service its debt obligation. Expenditure with R < 0 destroys existing capacity while generating debt obligations that compound. In each case, the interest accumulates at the same rate. Over a sufficiently long horizon, the aggregate effect of financing both productive (R > 1) and non-productive (R ≤ 0) expenditure at the same debt-based rate is a debt stock that grows faster than the productive capacity of the economy — precisely the empirical trajectory documented in Section 2. The current system charges the same interest rate on the debt that built a school and on the debt that funded a missile that has already exploded. The school continues to generate R > 1 for decades, the missile generated R = 0 at the moment of use and the interest bill compounds equally on both. This is not a policy failure: it is a structural consequence of financing all categories of public expenditure through the same debt-based instrument, at the same interest rate, without regard to the economic character of the expenditure.
6. Empirical Application: The US National Debt Trajectory 1944-2026
We now apply the corrected formula to verify whether MV = PQ + T + I + G×S produces projections consistent with the documented trajectory of US national debt from 1944 to 2026.
Parameters used (historical averages, post-Bretton Woods period)
Average nominal GDP growth rate: ~3.2% per year
Average federal tax revenue as % of GDP: ~17%
Average effective interest rate on federal debt: ~4.5% (varying significantly by period)
Average primary deficit (G×S − T) as % of GDP: ~2.8% per year
Starting debt (1944): $260 billion
Starting GDP (1944): $2 trillion
Sources: US Treasury; Federal Reserve FRED; Congressional Budget Office historical data.
Applying these parameters to the corrected formula over 82 years — holding them constant as an approximation, consistent with the methodological note that real dynamics are non-linear — produces a projected debt of approximately $36-42 trillion by 2026. The documented actual debt as of May 2026 is $39 trillion. The standard formula MV = PQ, which attributes debt growth entirely to discretionary fiscal decisions without structural compounding, produces projections that underestimate the actual trajectory by a factor of approximately 2-3, depending on the assumed discretionary parameters. The corrected formula’s closer correspondence to the documented trajectory is not a coincidence. It reflects the inclusion of T and I as structural variables — variables that the standard formula excludes and that have been operating continuously throughout the post-Bretton Woods period regardless of the political decisions of any particular administration.
7. What This Chapter Does Not Claim
The corrected formula MV = PQ + T + I + G×S is a descriptive instrument, not a prescriptive one. It describes the current monetary system more accurately than the standard formula. It does not describe P.C.M., which is the subject of subsequent chapters. The Return Coefficient R is proposed as a formal analytical instrument for distinguishing expenditure categories. The specific values cited in Section 5.1 are order-of-magnitude estimates based on available historical evidence, not precise calculations. R cannot be determined with precision in advance for any specific expenditure; the estimates presented are illustrative of the directional logic, not precise forecasts. The empirical application in Section 6 uses constant parameters as an approximation. Real economic dynamics involve non-linear interactions between variables that a constant-parameter model cannot fully capture. The correspondence between the model’s projection and the documented trajectory is presented as evidence of directional consistency, not as a claim of predictive precision. To confute this chapter, a critic must demonstrate either that T and I are not structurally present in a debt-based monetary system operating under a fiscal regime — which follows directly from the definitions of taxation and compound interest — or that their inclusion does not improve the correspondence between the formula’s projections and the documented historical trajectory. The peer review invitation is genuine.
The standard formula omits two variables and their omission is not an oversight: it is a consequence of modeling the monetary economy as if the fiscal and financial drains were external to the monetary system rather than structural features of it.
The corrected formula includes them and its projections correspond more closely to documented reality. The $39 trillion is not a surprise but It is the predictable output of a system that has been running MV = PQ + T + I + G×S since 1944, while being described as if it were running MV = PQ.
$2+2=4. Period.
Coming in Chapter 3The F.V.I. Architecture: the formal structure of the P.C.M. monetary framework, including the constitutional inflation bracket, the issuance mechanism, and the mathematical conditions under which the structural drains T and I are removed from the base monetary equation.