The Equality-Indexed Monetary-Trade System.

Reforming Global Money and Trade for Equality

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This paper develops the Equality-Indexed Monetary-Trade System (EIMTS) — a synthesis of mechanism design, distributional macroeconomics, and trade theory built around a single architectural claim: that the absence of any distributional term in the world's two most consequential rule-following institutions, the Taylor Rule and the gravity model of trade, is a design feature rather than a technical accident, and that an explicit equality instrument can be added to each without violating known impossibility results. The paper's contribution is integrative rather than paradigmatic. It does not claim a complete macroeconomic theory of inequality, nor a replacement for the existing heterogeneous-agent New Keynesian literature (Kaplan, Moll, & Violante, 2018; Auclert, 2019; Acharya, Challe, & Dogra, 2023). It claims, more narrowly, that an equality term in monetary and trade architecture is architecturally feasible and conditionally consequential, and that the conditions under which it matters are conditions our data strongly suggest already obtain.

The architectural feasibility result rests on the principal theorems. Theorem 1.1' establishes that any Taylor-form rule embeds distributional asymmetry as a design feature, with a conditional corollary (premised on r > g and incomplete fiscal offset) that the architectural asymmetry produces a structural drift in the Gini. Theorem 3.1' establishes bounded logistic convergence of the Gini to an asymptote Ḡ < 1 that depends on the strength of counterforces. Theorem 4.7' establishes an AGV-based incentive-compatible architecture for cross-national coordination, committing to BIC + IIR + ex-ante BB + ex-ante efficiency under explicit common-prior and quasi-linearity caveats. Theorem 5.4' establishes equilibrium existence under the Atkinson constraint, with the Bridging Proposition recovering Gini-denominated reporting. Theorem 5.6' establishes the second-welfare-theorem analogue as a non-operational existence result: the required initial-endowment redistribution exceeds the architecture's seigniorage capacity. Theorem 5.7'' establishes a conditional Kaldor-Hicks / aggregate-welfare improvement over the unconstrained baseline, supplying a potential-Pareto rather than actual-Pareto ordering since the supporting compensations are not operational under Theorem 5.6'.

The architecture's bounded tensions are stated affirmatively rather than as a catalogue of retreats. The target is disposable income Gini (WID and LIS standards); the wealth-channel side effect is acknowledged and routed to a macroprudential complementarity (LTV/DSTI caps, differential capital surcharges) that sits outside the monetary-trade core and is staged jointly with Phase II of the §7 transition path. The flagship national instrument addresses within-country inequality; the more pressing between-country diagnosis is relocated to the §7 multilateral coordination architecture (Bretton Woods 2.0, equality clearinghouse, OECD-equality-tax analogue). The AGV mechanism's BIC guarantee rests on a common-prior assumption that the IEMA (§7.2) is designed in part to support, and on a quasi-linearity domain (|t_i| ≤ 5% of GDP) that holds where transfers are too small to move the between-country diagnosis and degrades where they would matter — a structural tension the architecture surfaces rather than resolves. The §5.4 fair exchange-rate corridor is a non-binding monitoring band, with a binding upgrade reachable only under coordinated capital-flow management consistent with the Mundell-Fleming trilemma.

Empirical anchoring comes from three case studies — Bhutan's Gross National Happiness framework, New Zealand's Wellbeing Budget, and Mondragon Cooperative Corporation — each treated as a partial implementation of one EIMTS instrument under unfavorable conditions. The pre-specified econometric designs of §6.5 furnish the falsification framework as full implementations begin to appear; the calibration-robustness observation salvaged from the removed simulation pipeline confirms that the rank ordering of nations by equality index is robust to the WID-vs-survey methodological choice. The paper closes with a fifteen-year transition path organized in three phases, a limitations-first conclusion, and an affirmative coda summarizing what the architecture does deliver within its bounded acknowledged tensions: a formally specified, impossibility-result-consistent, institutionally stageable equality-indexed monetary-trade system.

Keywords: equality-indexed monetary policy, disposable income Gini, mechanism design, AGV mechanism, Atkinson constraint, gravity model, distributional macroeconomics, Piketty dynamics, HANK, Bayesian incentive compatibility, macroprudential complementarity, Mundell-Fleming trilemma, Kaldor-Hicks, transition path, Bretton Woods, OECD minimum tax

Revision Note

The paper's argument has been refined across four prior referee rounds; the v6 draft incorporates the v5 referee's "accept with minor-to-moderate revision" disposition. Briefly: v3 (the original 21,640-word draft) presented the architecture with several over-claims — DSIC + ex-post BB + Pareto-dominance combinations forbidden by Green-Laffont; Gini "approaches 1 in finite time"; binding fair-value exchange-rate corridor under open capital accounts. v4 applied Proof Repairs v2, primed Theorems 1.1', 3.1', 4.7', 5.4' to reflect the corrected commitments, moved the CGE formal specification into Appendix A, and softened §6.5 simulation magnitudes against the February 2026 audit. v5 confronted the v4 referee report by removing §6.5 simulation results entirely (the referee's binary choice between bug-disclaimed numbers and removal); committing the target object to disposable-income Gini with explicit wealth-channel acknowledgment and macroprudential complementarity; reframing §5.4 as a non-binding monitoring band consistent with Mundell-Fleming; dropping the DSIC strengthening of Theorem 4.7' because the five-dimensional EIMTS type space does not admit the Myerson one-dimensional payment identity without a defensible scalar projection; reconditioning Theorems 5.6' and 5.7' on operational and externality lower bounds; and surfacing the §3.7 within-/between-country target-instrument mismatch. v6 tightens the i/M independence accounting under endogenous money (new §5.2.x), corrects the Bridging Proposition Atkinson formula at general ε (the v5 formula coincides with the correct formula only at ε = 2), disambiguates the λ and μ notation that the v5 referee identified as carrying four and two distinct meanings respectively, surfaces the quasi-linearity / consequential-region tension between Assumption 4.A and the magnitudes required to move the between-country diagnosis, reclassifies Theorem 5.7'' as conditional Kaldor-Hicks rather than Pareto, adds a macroprudential row to the §5.1 Tinbergen instrument-target table, and closes the Conclusion with an affirmative coda stating what the architecture delivers within its bounded acknowledged tensions. v7 removes residual version-lineage commentary from the body chapters and tightens four notation issues that survived the prior round. v8 (this draft) reconciles the §6.4 Mondragon figures with the v7 referee's verified-current counts (approximately 70,500 worker-owners and €11.2 billion in 2024 industrial-and-distribution sales, with explicit acknowledgment of the 2022 ULMA / Orona departures that reduced the headline workforce from approximately 80,000) and adds the Mondragon Corporation (2024) and TU Lankide (2024) entries to the references.

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§ IThe Monetary Diagnosis

Modern central banking is organized around an objective function that is, by any honest reading, narrower than the institutional power it directs. The canonical formulation, due to Woodford (2003), specifies the central bank's loss function as

L = E_t Σ δ_L^i [π²_{t+i} + λ_y(y_{t+i} − y*_{t+i})²] (1.1)

where L is the loss to be minimized, π is inflation, y is the log of real output, y* is the log of potential output, δ_L is the central bank's discount factor (see the notation glossary in Appendix A.1), and λ_y is the relative weight on output stabilization. The expectation operator E captures the forward-looking character of optimal policy.

The first observation to make about (1.1) is the most consequential. The loss function contains no term for income distribution, wealth concentration, or any measure of equality. Distribution enters neither as a target variable nor as a constraint. This is not an oversight. It reflects a theoretical consensus, dating to Tinbergen's (1952) assignment rule, that monetary policy should target aggregate variables while fiscal policy addresses distribution. The consensus has remained intellectually dominant even as the empirical literature documenting monetary policy's distributional consequences has grown to substantial weight (Coibion, Gorodnichenko, Kueng, & Silvia, 2017; Auclert, 2019; Andersen, Johannesen, Jørgensen, & Peydró, 2023; Lenza & Slacalek, 2023). The operational translation of (1.1) into a policy rule is the Taylor Rule (Taylor, 1993):

i_t = r* + π_t + α(π_t − π*) + β(y_t − y*) (1.2)

which prescribes the nominal interest rate as a linear function of the inflation gap and the output gap. The rule's simplicity is its strength and its failure. It responds to two signals only — whether prices are rising too fast and whether the economy is producing below capacity — and its mechanism, adjusting the price of credit, has profoundly asymmetric effects across the wealth distribution.

1.1 Asymmetric Transmission

Partition the population into asset holders A with wealth share w_A and wage earners W with wealth share w_W, where w_A + w_W = 1. Under the Taylor Rule regime, ∂W_A/∂i < 0 and ∂W_W/∂i ≈ 0 when i_t decreases (loosening); ∂W_A/∂i > 0 and ∂W_W/∂i < 0 when i_t increases (tightening). Monetary loosening primarily increases asset values, benefiting A; tightening primarily reduces employment and wages, harming W. The transmission is structurally asymmetric: the gains of loosening accrue to capital, while the costs of tightening fall on labor. The mechanism is mechanical. When the central bank lowers i_t, the present value of future cash flows from assets rises by ΔPV = −D · Δi/(1+i)², where D is duration; this is a property of present-value discounting, not an empirical conjecture (Christiano, Eichenbaum, & Evans, 2005). Asset holders benefit directly. Credit expansion stimulates investment but reaches wage earners only through employment with a lag of six to eighteen months. When the central bank raises i_t to fight inflation, the employment channel contracts faster than asset prices adjust because of downward wage rigidity and hiring frictions. The asymmetry is the joint product of the mathematics of discounting and the empirical fact that asset holdings are concentrated — the top decile of U.S. households holds approximately 89% of equities and 75% of bonds (Federal Reserve, 2022).

1.2 The Cantillon Effect

Cantillon (1755) observed that newly created money does not enter the economy uniformly. Those who receive new money first can spend it before prices adjust, gaining real purchasing power at the expense of those who receive it last. Model money creation as diffusion through a financial network G = (V, E) where V is the set of economic agents and E represents financial relationships. The Cantillon distance d_C(i) of agent i is the shortest weighted path from the money creation source (central bank or commercial bank) to agent i in G. Under money creation ΔM > 0, the real wealth gain to agent i is a decreasing function of Cantillon distance. The mechanism operates through the time lag between money injection and price adjustment: if the general price level P(t) follows P(t) = P_0 + (P_1 − P_0)(1 − e^(−θt)), an agent receiving the new money at time τ(i) proportional to d_C(i) gains real purchasing power Δm_i/P(τ(i)) − Δm_i/P_1, which is positive for agents close to the source and approximately zero for those far away.

This is a topological property of money-injection networks, not an empirical conjecture about institutions. It requires only that money enters through specific nodes, that the network has non-zero diameter, and that price adjustment is not instantaneous — all three conditions satisfied by the construction of the contemporary monetary system. Empirically, commercial banks (d_C = 1), financial institutions (d_C = 2), and corporate borrowers (d_C = 3) systematically capture more real value from money creation than wage earners (d_C ≥ 5) and transfer recipients (d_C ≥ 7).

1.3 Quantitative Easing as Empirical Demonstration

The Cantillon effect moved from theoretical concern to empirical demonstration with quantitative easing. Between 2008 and 2022 the Federal Reserve expanded its balance sheet from roughly $900 billion to $8.9 trillion — a ten-fold increase. The wealth effect of large-scale asset purchases (LSAPs) follows ΔW_i = α_i · ΔP(assets) + (1 − α_i) · Δw_i(labor), where α_i is agent i's asset share of total wealth. Because LSAPs operate primarily through the asset price channel and because α is highly concentrated (α > 0.9 for the top decile; α < 0.1 for the bottom half), the distributional consequence is mechanical: ΔGini(QE) > 0. Over the QE era the S&P 500 rose 582% from its March 2009 low to its January 2022 peak; over the same period median real wages rose approximately 8.7%. The asset-price channel of QE delivered roughly thirty trillion dollars in wealth gains predominantly to households already in the top decile, while wage gains were modest and broadly distributed (Saiki & Frost, 2014; Mumtaz & Theophilopoulou, 2017; Furceri, Loungani, & Zdzienicka, 2018). The HANK literature (Kaplan, Moll, & Violante, 2018; Auclert, 2019) has refined the formal mechanism through which this happens, distinguishing the direct interest-rate channel, the indirect general-equilibrium channel, and the asset-revaluation channel that dominates for portfolio-rich households; the qualitative direction is no longer in serious dispute among researchers working in this literature.

1.4 Seigniorage and Its Distribution

Seigniorage — the revenue from money creation — is a hidden fiscal instrument with first-order distributional consequences. Under the current architecture the vast majority of money is created not by central banks but by commercial banks through lending. The money-multiplier decomposition S = S^cb + S^bank with S^bank = (m − 1)ΔM_0/P that older textbook accounts used is theoretically obsolete. The contemporary central-banking consensus, articulated explicitly in the Bank of England's Money creation in the modern economy report (McLeay, Radia, & Thomas, 2014) and developed further in the modern-monetary-theory literature (Wray, 2015; Kelton, 2020), is that commercial-bank money creation operates endogenously through the loan-issuance process — banks create deposits when they extend credit, constrained by capital requirements, profitability of lending, and central-bank policy rates rather than by a fixed multiplier on reserves. The seigniorage that accrues to the commercial-banking sector is the interest-rate spread between lending and deposit rates, sized by the credit demand the banking sector chooses to accommodate at prevailing margins; it is not mechanically determined by the central bank's reserve injection.

The distributional consequence is unchanged by the corrected framing: commercial-bank credit creation systematically advantages the agents closest to the credit-creation point (the Cantillon distance gradient of §1.2), and the asset-revaluation channel of §1.1 directs the gains predominantly to the top of the wealth distribution. What the endogenous-money framing changes is the architectural reading of the policy-instrument space: the binding constraint on money creation by sovereign currency issuers is inflation, not a multiplier-and-reserves accounting identity (Kelton, 2020), so an equality-indexed money supply rule M(eq) = M(base) · Ψ(E_t) is feasible provided the inflation constraint is respected. The Tinbergen separation that traditionally assigned distributional response to fiscal policy alone presupposes precisely the multiplier-framed mechanical accounting that the endogenous-money literature has displaced; once distribution can be addressed through the price and quantity of credit at margins the central bank actually controls, the conventional assignment is a design choice rather than a technical constraint. The implications of this commitment for the operational independence of the equality-indexed money-supply rule (§5.2) are addressed in §5.2.x.

1.5 Theorem 1.1': Monetary Design Embeds Distributional Asymmetry

The four mechanisms above — asymmetric transmission, Cantillon ordering, the QE asset-price channel, seigniorage allocation — share a common structural feature: they hold for any parameterization of the Taylor Rule because the rule's action space does not include a distributional instrument. The architectural part of the result holds by inspection; the consequential part — that the Gini drifts upward under current-architecture parameterizations — is conditional on empirically testable premises (r > g; incomplete fiscal offset) rather than a structural axiom. The theorem statement below separates the two.

Theorem 1.1' (Monetary Design Embeds Distributional Asymmetry). Under any monetary policy rule of the form i_t = f(π_t, y_t) that excludes distributional terms, the following hold: (a) The transmission mechanism is asymmetric: rate changes affect asset holders and wage earners differently in both sign and magnitude. (b) The money creation process is sequential: agents closer to the point of injection capture real purchasing power before price adjustment. (c) These asymmetries are design features of the policy architecture — they hold for any parameterization of α and β in the Taylor Rule — not consequences of particular economic conditions.

Conditional Corollary. If additionally (i) r > g holds on average over the relevant horizon, and (ii) fiscal redistribution is insufficient to fully offset channels (a)–(b), then the Gini coefficient increases over time. The rate of increase is bounded below by a function of the Cantillon distance gradient and the r − g gap.

Part (a) follows from present-value mechanics combined with the empirical concentration of asset holdings. Part (b) is the topological property of money-injection networks established above. Part (c) is by inspection: the Taylor Rule i_t = r* + α(π_t − π*) + β(y_t − y*) responds exclusively to aggregate variables, and no parameterization can introduce a distributional response without enlarging the rule's domain. The Conditional Corollary names the two empirical premises explicitly. Premise C1, r > g, is Piketty's (2014) central finding, documented across two centuries of data for twenty-plus countries; it is an empirical regularity, not a theorem, and it has been challenged in the live literature by Mankiw (2015) and Krusell and Smith (2015), whose objections rest on the claim that consumption out of capital income should narrow the gap. Both Piketty's reply and the subsequent calibration work suggest the gap persists under most reasonable parameter choices; we treat C1 as a maintained hypothesis. Premise C2, incomplete fiscal redistribution, is captured by the offset ratio φ = |ΔGini(fiscal)| / |ΔGini(monetary)|, which Causa and Hermansen (2020) estimate at 0.3–0.6 for OECD economies. Full proof is in Appendix A.4.

The qualitative consequence of 1.1' is sharper even though the quantitative claim is weaker. The architectural part is now unconditional: no Taylor-Rule parameterization will respond to distributional outcomes, because the rule's action space does not include them. The conditional part names the testable premises under which the architectural feature becomes a structural Gini drift, rather than smuggling those premises in unannounced. This is the necessity, under the conditional corollary, of an explicit distributional instrument in the policy rule — not necessarily the γ(G_t − G*) term we propose in Chapter 5, but some instrument that enlarges the rule's domain to include distribution. The case for the specific γ(G_t − G*) augmentation rests on the equilibrium and welfare results of Chapter 5; the architectural case for any distributional instrument rests on Theorem 1.1' alone.

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§ IIThe Trade Diagnosis

If the monetary system embeds distributional asymmetry through silence, the trade system embeds it through a richer set of mechanisms — none individually decisive, but jointly producing a structural pattern that the post-1945 trade-liberalization consensus has consistently underestimated. This chapter walks through the canonical trade-theoretic apparatus — Heckscher-Ohlin, Stolper-Samuelson, Prebisch-Singer, new trade theory, the gravity model — not because each remains the live frontier of the field (it does not; modern trade theory has moved on to Eaton-Kortum, Melitz, and the global-value-chains literature of Antràs, 2003, 2020) but because each contributes one element of the inequality-generating mechanism the Equality-Indexed Monetary-Trade System is designed to interrupt. The exposition is therefore historical in sequence and synthetic in argument: we trace the structure that emerged across these literatures, locate the assumption failures that produce the inequality pattern, and conclude with a theorem (Theorem 2.4') that the architecture transfers real surplus from periphery to core under joint conditions overwhelmingly satisfied in the empirical record.

2.1 The Heckscher-Ohlin Framework and Its Limitations

The Heckscher-Ohlin (H-O) model, developed by Heckscher (1919) and Ohlin (1933), provides the canonical comparative-advantage account: countries export goods that are intensive in their abundant factors and import goods intensive in their scarce factors. Two countries (Home labor-abundant, Foreign capital-abundant), two factors (labor L and capital K), two goods (manufactures M and primary commodities P); each country has factor endowments differing in ratio, common technology coefficients a_ij representing the amount of factor i required per unit of good j, and access to costless trade. Under these assumptions the labor-abundant country exports labor-intensive goods, the capital-abundant country exports capital-intensive goods, and a striking corollary — the Factor Price Equalization theorem (Samuelson, 1948) — predicts that trade in goods alone causes factor prices to converge across countries. Workers in labor-abundant countries need not migrate to capital-abundant countries because trade brings them the same real wage benefit.

Six decades of trade liberalization have not produced factor price equalization. Real wages in high-income countries remain 5–15 times higher than in low-income countries despite continuous integration. The most fundamental failure is in technology. Technology is not identical across countries; high-income countries employ vastly more capital-intensive and sophisticated production processes than low-income countries. A manufacturing facility in Germany producing precision machinery uses more than ten times the capital per worker of a comparable facility in Bangladesh producing textile machinery. The capital-efficiency gap Ω_M = a_KM^F / a_KM^H > 1 implies that wage-rental ratios cannot equalize: the foreign (low-income) economy must employ less efficient processes to remain competitive, locking wages permanently below the capital-rich country's level. Trade equilibrium does not lead to FPE; it locks countries into their technological positions. Costinot and Rodríguez-Clare (2014) and Eaton and Kortum (2002) provide the modern quantitative trade theory in which the FPE failure is no longer a puzzle but a property — productivity differences are the primary observable, and the framework predicts persistent wage gaps as a consequence.

The second failure is in transport costs, which are far from zero and asymmetric. Primary commodities face ad valorem transport costs of roughly 8–15% of value; manufactures face 2–5%. The arbitrage-free pricing condition becomes P_j^F = P_j^H · (1 + τ_j) for importers; the periphery faces systematically higher prices for imported capital-intensive manufactures while receiving lower prices for primary-commodity exports. The terms of trade move against the commodity exporter. The third failure is incomplete specialization: empirically, countries do not produce both goods in equilibrium at all factor-endowment combinations, particularly at the lower end of the development distribution where commodity specialization is near-complete. When H-O.1 and H-O.2 are violated jointly, trade liberalization can increase within-country inequality in the labor-abundant country even when it increases average real incomes — by creating low-wage commodity-sector employment while leaving the high-wage manufacturing sector inaccessible because of the technology gap.

2.2 Stolper-Samuelson and Within-Country Inequality

While H-O explains which goods countries export, Stolper and Samuelson (1941) explain the distributional consequences. In a two-factor, two-good model, an increase in the price of the capital-intensive good causes the wage (return to labor) to fall proportionally more than the price rises, while the rental rate (return to capital) rises proportionally more than the price rise. This is the magnification effect: factor returns are magnified relative to goods prices. Trade liberalization in a capital-abundant country opening to trade raises the price of the capital-intensive export good (this is the H-O export pattern), so capital returns r rise more than proportionally while labor returns w fall — a clarifier worth stating explicitly because the direction of the result is sometimes misreported in policy discussion. The Stolper-Samuelson direction depends on which factor is used intensively in the good whose relative price rose; "rise" and "fall" reverse across countries depending on the country's factor endowment.

The practical implication is stark. A 10% fall in the relative price of manufactures (in a country where manufactures are the capital-intensive sector) reduces wages by more than 10% while increasing capital returns by more than 10%. When trade liberalization causes the relative price of primary commodities to fall — which it persistently does, as the Prebisch-Singer literature established — the Stolper-Samuelson mechanism predicts increasing wage inequality over time. Within the periphery (primary-commodity exporters) this manifests as a widening wage gap between skilled workers in protected capital-intensive sectors and unskilled workers in commodity production. Autor, Dorn, and Hanson's (2013) China-syndrome analysis documents the comparable mechanism in the core: import competition concentrated in particular regional labor markets produced persistent local wage and employment losses that the standard aggregate gains-from-trade calculation did not capture. Stolper-Samuelson is not an inequality result against trade per se; it is a result that trade liberalization redistributes within both partners, and the redistribution favors capital where commodity prices fall and labor where commodity prices rise. In the world we actually have, with persistent commodity-price decline, the net effect is upward inequality drift in both the periphery and the import-competing regions of the core.

2.3 Prebisch-Singer: Structural Terms-of-Trade Deterioration

Prebisch (1950) and Singer (1950), working independently, observed that primary-commodity exporters experienced persistent deterioration in their terms of trade. Their hypothesis — that commodity prices decline secularly relative to manufactured-goods prices — fundamentally challenges the symmetric-gains-from-trade reading. Two mechanisms drive the secular deterioration. Income elasticity divergence: demand for primary commodities is income-inelastic (∂Q_primary/∂Y < 1) while demand for manufactures is income-elastic (∂Q_manuf/∂Y > 1). If global real income grows at rate g, commodity demand grows at g · η_primary < g and manufactured-goods demand at g · η_manuf > g. With supply of commodities relatively fixed by land and resource constraints, the slow-growing demand produces persistent price pressure. Technological progress in manufacturing: manufacturing sectors in developed countries experience rapid productivity growth (φ_m > 0.03 per year empirically), reducing manufactures' supply price, while primary-commodity sectors face slower productivity growth (φ_p < 0.02 per year) and binding environmental constraints. The productivity gap drives manufacturing costs down faster than commodity costs, reinforcing the price decline.

Combining mechanisms, the real terms of trade for a commodity exporter follows ToT(t) = ToT(0) · exp(−λ_ToT · t) with λ_ToT ≈ 0.01–0.02 per annum empirically (subscripted to distinguish from the four λ-uses disambiguated in §A.1; this is a §2.3-local terms-of-trade decline rate, not one of those four). Over the sixty-year window 1960–2020, this implies ToT(2020) / ToT(1960) ≈ 0.41 — the terms of trade of a typical commodity exporter in 2020 are roughly 41% of their 1960 level. UNCTAD's continuous dataset (United Nations Conference on Trade and Development, 2023) confirms the pattern: for non-fuel commodity exporters the real price index fell from 100 (1960) to approximately 42 (2023), a decline of 58% in real terms. Countries that diversified away from commodities — South Korea, Taiwan, Singapore — achieved convergence in living standards; those that remained commodity-dependent stagnated. The Prebisch-Singer hypothesis is not merely theoretical: it describes observable, persistent global inequality in trade patterns over the full post-1945 record.

2.4 New Trade Theory: Self-Reinforcing Advantages

Classical trade theory assumes perfect competition and constant returns to scale. Krugman's (1979, 1991) work on increasing returns and monopolistic competition demonstrated that trade advantages can be self-reinforcing and entirely path-dependent. A production process exhibits increasing returns when average costs fall as production quantity increases: AC_i = F / Q + c, where F is a fixed cost. As Q rises, AC falls. This creates economies of agglomeration: regions with larger concentrations of a sector benefit from lower costs, attracting more firms, which further reduces costs. The core-periphery model that results predicts that the region achieving manufacturing concentration first — the core — retains its advantage indefinitely, even after initial technological or endowment differences disappear. The steady-state wage in each region depends on market potential, the sum of nearby demand weighted by trade costs. The core, with a large local market and low internal trade costs, achieves higher wages; the periphery, with small local markets and high trade costs to large markets, cannot match them.

Krugman (1980) further showed that larger markets specialize in goods with increasing returns — the home market effect. This explains why the United States produces a disproportionate share of high-tech goods and services not because of endowments but because of market size, while peripheral countries with small local markets cannot achieve the scale needed to compete in increasing-returns sectors. They specialize in commodities by elimination, not by comparative advantage in any deep sense. Melitz (2003) and Helpman (2011) refined this further: heterogeneity in firm productivity means that trade liberalization produces winners and losers within sectors, reallocating production toward the most productive firms in each country. The mechanism explains why trade liberalization has failed to produce convergence and why aggregate gains-from-trade calculations can be technically correct while masking very large distributional consequences in both partner countries.

2.5 The Gravity Model

The gravity model of trade is the workhorse empirical framework in international economics. Anderson and van Wincoop (2003) provide the theoretically-grounded specification reconciling gravity with new trade theory:

T_ij = (Y_i · Y_j / Y_world) · (t_ij / (Π_i · P_j))^(1−σ) (2.27)

where T_ij is bilateral trade flow, Y_i and Y_j are GDPs, t_ij is bilateral trade costs (iceberg costs: selling one unit requires shipping 1+t_ij), Π_i and P_j are multilateral resistance indices, and σ is the elasticity of substitution (typically 5–8). The multilateral resistance terms capture each country's position in the global trade network: a country with many costly trade relationships faces high multilateral resistance and thus trades less with all partners. Head and Mayer (2014) survey the workhorse-toolkit-cookbook role gravity plays in modern empirical trade work. Empirical gravity estimates reveal systematic asymmetries: high-income country exporters face lower effective trade costs than low-income country exporters for identical products, reflecting quality standards, logistics networks, financial access, and informational advantages correlated with income. The model as currently estimated includes t_ij as an exogenous parameter capturing geography, tariffs, and distance. The actual trade costs facing exporters depend on income level, institutional quality, and technological sophistication — factors that create asymmetric disadvantages for periphery countries that the canonical gravity specification absorbs into the multilateral resistance term without exposing as policy-relevant.

The Equality-Indexed Monetary-Trade System modifies this baseline by introducing an equality index E_ij that adjusts trade costs as a function of the wage gap between exporters and importers. The modified gravity equation is

T_ij^EIMTS = (Y_i · Y_j / Y_world) · (t_ij · f(E_ij) / (Π_i · P_j))^(1−σ) (2.30)

where f(E_ij) is decreasing in the wage gap and equals 1 under current conditions. The full development of this modification — and its formal elasticity ζ (renamed from γ to avoid overloading with the Taylor Rule equality coefficient; see Appendix A.1) — appears in Chapter 5.

2.6 Theorem 2.4': Conditional Trade-Structural Inequality

We can now state the trade-side analogue to Theorem 1.1'. The five mechanisms developed in this chapter — technological asymmetry (2.1), Stolper-Samuelson wage compression in commodity exporters (2.2), Prebisch-Singer terms-of-trade deterioration (2.3), new-trade-theory core-periphery lock-in (2.4), and gravity-model multilateral-resistance asymmetry (2.5) — operate independently but do not unconditionally point in the same direction. Each admits counter-cases in isolation. Their joint operation under the conjunction of four enabling conditions, identified explicitly below, produces the structural-bias pattern that the post-1945 trade-liberalization consensus has consistently underestimated.

Theorem 2.4' (Conditional Trade-Structural Inequality). Under the joint conditions (i) divergent factor endowments across countries, (ii) heterogeneous price elasticities in primary versus industrial export sectors, (iii) self-reinforcing agglomeration economies in manufacturing, and (iv) gravity-weighted bilateral trade flows reflecting unequal multilateral resistance, the five mechanisms summarized in §§2.1–2.5 — Heckscher-Ohlin factor specialization, Stolper-Samuelson wage compression in commodity exporters, Prebisch-Singer secular terms-of-trade deterioration, new-trade-theory core-periphery lock-in, and gravity-model multilateral-resistance asymmetry — jointly produce a structurally biased distribution of gains from trade. The net real income gain to a peripheral country from trade satisfies ΔY_peripheral/Y_peripheral < ΔY_core/Y_core over multi-decade periods, and the absolute wage gap d(w_peripheral − w_core)/dt < 0 widens over time, by joint conditional inference from the five mechanisms.

The architectural reading mirrors 1.1'. The four joint conditions are structural features of the current trade system, not contingent properties that depend on particular policy choices in peripheral countries. As with the monetary case, the consequential reading — actual deterioration in observed terms of trade and wage gaps — is conditional on the empirical persistence of all four conditions jointly, each of which is observably persistent at the multi-decade scale. No policy adjustment within the peripheral country alone can overcome the pattern unless the architecture itself is reformed: this is the trade analogue of the necessity, under Theorem 1.1', of an explicit distributional instrument in the policy rule. Full proof structure follows the conditional template of 1.1' and appears in Appendix A.5.

The corollary is now familiar. The convergence of global real wages requires modification of the trade architecture itself — specifically, adjustment of multilateral trade costs and gravity-model parameters to internalize an equality objective. The EIMTS framework provides this by introducing E_ij into the gravity equation and by routing the resulting trade-cost adjustments through the monetary side via the fair-value exchange-rate monitoring band developed in §5.4. The detail of the architecture is the work of Chapters 4 and 5.

2.7 Synthesis

This chapter has established that contemporary trade architecture transfers real surplus from low-wage to high-wage economies under five mutually reinforcing conditions all observed in the empirical record. The H-O model's assumptions of identical technology and zero transport costs systematically fail; the Stolper-Samuelson mechanism redistributes within both partners and produces widening wage inequality where commodity prices fall; Prebisch-Singer documents the secular deterioration of primary-commodity terms of trade over six decades of UNCTAD data; new trade theory shows that manufacturing advantages are self-reinforcing and path-dependent rather than endowment-based; the gravity model reveals how trade costs asymmetrically burden low-income exporters. Theorem 2.4' synthesizes these into the conditional architectural claim, and draws the consequential claim from observably persistent empirical conditions. The trade system does not fail because of mismanagement in peripheral countries. It fails because the design assumptions on which liberal trade policy was justified are not, and have never been, satisfied in the world it was implemented in. The case for equality-adjusted trade weights — the formal subject of Chapter 5 — is structurally identical to the case for an equality term in the policy rule: the rule's action space, as currently constructed, does not respond to the distributional consequences its own mechanism generates.

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§ IIIThe Inequality Mathematics

The two preceding chapters established that contemporary monetary and trade architectures embed distributional asymmetry through the absence, in each rule's action space, of any term that responds to distributional outcomes. This chapter develops the formal machinery required to talk precisely about distributional outcomes: the Gini coefficient and Lorenz curve, the Atkinson index (which will become the formally binding constraint in our model — see §5.6 and Theorem 5.4'), the Theil index and its decomposability, Piketty's two fundamental laws, and the Pareto-tail dynamics that emerge endogenously from multiplicative growth. The chapter then synthesizes these into Theorem 3.1', the revised inequality-monotonicity result, and corollaries that explain why the EIMTS constraint matters once bounded convergence (rather than finite-time approach to 1) is the operative dynamics.

3.1 The Gini Coefficient: Derivation and Properties

The Gini coefficient, formulated by Gini (1912), is the most widely-used scalar measure of inequality. For a population with incomes y_1, y_2, …, y_n sorted in ascending order with mean μ, the mean absolute deviation is MAD = (1/n²) Σ_i Σ_j |y_i − y_j|. The Gini coefficient normalizes by 2μ:

G = MAD / (2μ) = (1/2nμ) Σ_i Σ_j |y_i − y_j| (3.2)

Rearranging into the standard single-index form yields G = (2 Σ_i i·y_i) / (n · Σ_i y_i) − (n+1)/n. The Gini ranges from 0 (perfect equality) to 1 (perfect inequality). Define the Lorenz curve L(p) as the fraction of total income earned by the poorest fraction p of the population; L rises from (0,0) to (1,1), and perfect equality is the line L(p) = p. The Gini coefficient equals twice the area between the Lorenz curve and the equality line: G = 1 − 2 ∫_0^1 L(p) dp. The Gini is bounded in [0,1], satisfies the Pigou-Dalton transfer principle (a progressive transfer strictly decreases G), is population-invariant, and is scale-invariant.

It has two limitations that matter for our purposes. First, different income distributions can yield identical Gini values while differing significantly in social welfare implications — the Gini is insensitive to whether inequality comes from many moderate differences or from a small ultra-wealthy elite. Second, unlike some inequality measures, the Gini is not decomposable into within-group and between-group components, limiting its usefulness for the multi-level monitoring the EIMTS architecture requires. Both limitations motivate the introduction of the Atkinson and Theil indices below; the Atkinson, in particular, will become the formally binding constraint in the equilibrium model, with the Gini surviving as a reporting metric via the Bridging Proposition of §5.6.

3.2 The Atkinson Index

Atkinson (1970) developed an inequality measure grounded explicitly in social welfare theory. For a distribution with mean μ, the Atkinson index with inequality-aversion parameter ε is

A^ε(x) = 1 − [Σ_i (x_i/μ)^(1−ε) / n]^(1/(1−ε)) (3.6)

with the limiting log-form at ε = 1. The parameter ε ∈ (0, ∞) reflects society's aversion to inequality: ε → 0 makes A^ε insensitive to inequality and focuses on the mean alone; ε → ∞ approaches a maximin (Rawlsian) criterion concentrating on the poorest individual. The welfare interpretation is the fraction of income society could sacrifice while maintaining social welfare if all income were distributed equally. The equally-distributed-equivalent income is Y_ede = μ · [Σ_i (x_i/μ)^(1−ε) / n]^(1/(1−ε)), and A^ε = 1 − Y_ede/μ.

The Atkinson index has three properties that the Gini lacks and that will matter for the equilibrium-existence proof in §5.6. First, it is grounded in social welfare theory and admits democratic determination of ε rather than treating inequality as a purely technical matter. Second, for ε > 1 the set {x : A^ε(x) ≤ A*} is convex — a property the level sets of the Gini do not in general possess. This is what makes Atkinson the natural formal constraint for the EIMTS equilibrium problem; the model carries the Atkinson constraint A^ε(x) ≤ A* throughout, with Gini-denominated reporting recovered via the Bridging Proposition. Third, the Atkinson index is in the Lorenz-respecting family of inequality measures, so it agrees with the Gini ordering on the empirically relevant Lorenz-comparable subset of distributions.

3.3 The Theil Index and Decomposability

The Gini's non-decomposability is a critical limitation for a multi-level architecture. The Theil index, derived from information theory (Theil, 1967), provides the decomposable alternative. For a distribution with mean μ, the Theil T-statistic is T = (1/n) Σ_i (y_i/μ) ln(y_i/μ), which equals zero under perfect equality and ranges up to ln(n). When the population is partitioned into K groups with sizes n_k, means y_k, and income shares s_k = n_k y_k / (nμ), the Theil decomposes additively as

T_total = T_between + T_within (3.9)

where T_between = Σ_k s_k ln(y_k/μ) and T_within = Σ_k s_k · T_k. The decomposition follows from ln(y_i/μ) = ln(y_i/y_k) + ln(y_k/μ) and regrouping. Decomposability lets EIMTS monitor equality simultaneously at multiple levels — within nations, between nations, and globally — and lets the architecture respond to each level with its appropriate instrument. The Theil is a member of the generalized-entropy family GE(α) = (1/(α² − α)) · [(1/n) Σ_i (y_i/μ)^α − 1]; α = 0 yields mean logarithmic deviation, α = 1 yields Theil T, α = 2 yields Theil L. Different α values emphasize different parts of the distribution. Cowell (2011) provides the standard reference.

3.4 Piketty's Fundamental Laws

Piketty's (2014) synthesis of two centuries of economic data establishes two fundamental laws governing capital accumulation. Let κ denote the capital-to-income ratio (see Appendix A.1 for the notation glossary), r the rate of return on capital, and α the capital share of total income. Piketty's First Law is the accounting identity α = r · κ. Piketty's Second Law states that in the long run, the capital-income ratio converges to κ* = s / g, where s is the savings rate and g is the economic growth rate. In steady state, capital accumulates at rate s per year and growth erodes its share at rate g per year, yielding a stock-to-flow ratio of s/g. Combining the two laws, α = r · s / g.

The critical observation: empirically and historically, r > g. Capital returns of roughly 4–5% historically exceed economic growth of 1–2%, so α rises over time. Wealth for a capital owner with initial wealth W_0 grows as W_t = W_0(1 + r)^t (ignoring consumption), while average wealth grows at (1 + g)^t. The compounding gap is substantial: over thirty years, an annual difference of 3% (r − g = 0.03) compounds to a 27.1% larger relative position for capital. Mankiw (2015) and Krusell and Smith (2015) have contested the inevitability of this gap — Mankiw argues that consumption out of capital income should narrow it; Krusell and Smith argue that depreciation in net-of-depreciation accounting changes the picture substantially. Piketty's reply, supported by subsequent calibration work, is that even under reasonable consumption-and-depreciation accounting the gap persists in most periods, and the post-WWII compression — when r ≈ g for several decades — was a function of high growth, capital destruction, and confiscatory wartime taxation rather than a structural property of the economy. The debate continues; we treat r > g as a maintained hypothesis throughout, as in the proof of Theorem 1.1'.

3.5 Power-Law Wealth Distributions

Historical and contemporary wealth distributions do not follow bell curves. They exhibit heavy right tails described by power laws. The Pareto distribution P(X > x) = (x_m / x)^α (with x_m the minimum wealth level and α the shape parameter) is the canonical form; low α implies heavy tails (more ultra-wealthy), high α implies thinner tails. For a pure Pareto distribution, the Gini coefficient is G_Pareto = 1 / (2α − 1). With α = 1.5 — the empirically observed value for the top 1% in wealthy nations — the Gini contribution from wealth concentration alone is G = 1/2 = 0.50. Empirical evidence (Piketty & Saez, 2003; Milanovic, 2016; Piketty, Saez, & Zucman, 2018; Chancel, Piketty, Saez, & Zucman, 2022) gives top-1% tail exponents of α ≈ 1.5–1.7, top-0.1% exponents of α ≈ 1.0–1.2, and top-10% exponents of α ≈ 2.0–2.5, consistent across France, Germany, the UK, and the United States. The World Inequality Database, on which we will rely heavily for the case studies in Chapter 6 (Alvaredo, Chancel, Piketty, Saez, & Zucman, 2018; Chancel et al., 2022), confirms the persistence of these patterns at the longest time scales modern data permit.

Why do power laws emerge? Gabaix (1999) proved that they arise endogenously from multiplicative random growth. If agents have wealth W_i(t+1) = R_i(t) · W_i(t) with R_i drawn independently from a distribution with mean μ_r > 1, subject to non-negativity (no bankruptcy), the process generates a stationary distribution with a power-law tail. The tail exponent depends on the variance of growth rates: small differences in annual returns, compounded over decades, generate exponential differences in final wealth. An agent with consistent 6% returns versus 4% over forty years ends with (1.06/1.04)^40 ≈ 3.3× more wealth. Multiplicative processes naturally produce concentration. The Gini coefficient under such dynamics approaches a deterministic limit that depends on growth-rate variance. Solt's (2020) standardization work shows that across countries and decades, market Ginis cluster in a narrow band consistent with this prediction, with redistribution moving observed (disposable-income) Ginis below the market values by an amount that varies with the redistribution capacity of the state.

3.6 Theorem 3.1': Conditional Inequality Divergence

The Gini dynamics under the current monetary-trade architecture are monotonically increasing under the maintained empirical premises but converge to a bounded steady state Ḡ < 1 rather than reaching full concentration in finite time. Once counterforces are quantified, the underlying ODE delivers bounded logistic convergence; asymptotic convergence to Ḡ replaces the earlier finite-time-to-1 formulation.

Theorem 3.1' (Conditional Inequality Divergence). Under the maintained hypotheses (H1) r > g on average, (H2) current monetary architecture (no distributional term in the policy rule), and (H3) multiplicative income growth with stochastic shocks, the Gini coefficient satisfies dG/dt ≥ ε(r, g, φ, σ²) > 0, where ε is a lower bound depending on the r − g gap, the fiscal offset ratio φ, and the variance σ² of multiplicative shocks. The Gini trajectory is bounded above by a logistic function

G(t) ≤ Ḡ / (1 + ((Ḡ/G_0) − 1)e^(−εt))

where Ḡ < 1 is the upper bound reflecting natural counterforces. G(t) converges to Ḡ asymptotically but does not reach 1 in finite time.

The full proof appears in Appendix A.4. The sketch is as follows. Three inequality-increasing forces and three counterforces appear in the Gini dynamics. The forcing terms: the Piketty channel ΔG_1 = η_1(r − g)π_k where π_k is the capital income share; the monetary channel ΔG_2 = (1 − φ)η_2 aggregating the four sub-channels of Theorem 1.1'; the stochastic channel ΔG_3 = η_3 σ². The counterforces: progressive taxation reduces Gini by ΔG_4 = −τ̄ G; inheritance dilution contributes ΔG_5 = −δ G; entrepreneurial disruption contributes ΔG_6 = −μ_d(G − G_L), where μ_d is the entrepreneurial-disruption rate (see Appendix A.1 for the notation glossary). Combining yields a first-order linear ODE dG/dt = A − BG with A = η_1(r − g)π_k + (1 − φ)η_2 + η_3 σ² + μ_d G_L and B = τ̄ + δ + μ_d. The steady state is Ḡ = A/B. The solution to dG/dt = A − BG with G(0) = G_0 is G(t) = Ḡ − (Ḡ − G_0)e^(−Bt). When A/B > G_0 the Gini increases monotonically toward Ḡ; the convergence rate is B and the half-life is ln(2)/B.

Calibration under OECD-average parameters (τ̄ ≈ 0.03, δ ≈ 0.01, μ_d ≈ 0.02 in addition to the forcing parameters above) yields A ≈ 0.020, B ≈ 0.06, Ḡ ≈ 0.33 — meaning that from current OECD levels of inequality, the unconditional prediction would be modest convergence downward. Under global parameters without coordinated redistribution (τ̄ ≈ 0.005, δ ≈ 0.005, μ_d ≈ 0.01), B ≈ 0.02 and Ḡ ≈ 0.75–0.85, with half-life of roughly thirty-five years from current global between-country Gini levels (G_0 ≈ 0.70) toward Ḡ ≈ 0.80. Ḡ < 1 for any non-zero counterforce, so the Gini does not approach 1 in finite time; it converges asymptotically to Ḡ ∈ (0, 1). Under global conditions without coordinated redistribution Ḡ ∈ [0.75, 0.85] — severe but not complete concentration.

3.7 Why EIMTS Rather Than Coordinated Fiscal Redistribution

The OECD calibration above raises an obvious objection: if existing redistribution would, under OECD-average parameters, drive Ḡ to around 0.33, then the inequality problem is one of insufficient international coordination of existing tools rather than a fundamental architectural failure requiring a novel monetary-trade system. The objection deserves a direct answer; it is the most important objection to the case made in this chapter and the one that determines whether what follows is a contribution or a redundancy.

The answer has three parts. First, the coordination problem is the architectural problem. Existing fiscal redistribution works within nation-states because domestic tax authorities have enforcement power. International fiscal coordination — a global wealth tax, coordinated minimum corporate rates, anything cross-border at the scale required — requires sustained voluntary cooperation among sovereign nations with heterogeneous preferences and asymmetric incentives, precisely the setting where the mechanism design literature predicts cooperation will be fragile. The historical record confirms this: the OECD Base Erosion and Profit Shifting (BEPS) framework took over a decade to negotiate and remains incompletely implemented; the EU's financial transaction tax proposal (2011) has stalled indefinitely; actual cross-border fiscal redistribution (EU structural funds, the cumulative Marshall Plan analogue) amounts to roughly 1% of recipient GDP, an order of magnitude below what the model requires for Ḡ < 0.40 at global scale. The OECD-minimum-tax agreement of 2021 is the closest the international system has come to genuine fiscal coordination on inequality-relevant ground, and even its 15% rate represents a fraction of what serious global redistribution would entail.

Second, EIMTS is self-enforcing by construction. The AGV mechanism established in Theorem 4.7' (next chapter) provides Bayesian incentive compatibility: truthful participation is a Bayesian Nash equilibrium given a nation's beliefs about others' behavior. This is qualitatively different from a voluntary tax-coordination agreement, which is sustained only by repeated-game reputation effects and is vulnerable to unilateral defection whenever short-term gains exceed the discounted value of future cooperation (the fragility documented in folk-theorem analysis under imperfect monitoring). Third, EIMTS operates through the monetary channel, which has unique properties: central banks adjust interest rates continuously and with minimal legislative process, while fiscal redistribution requires parliamentary approval in each jurisdiction. The monetary channel's speed and automaticity make it complementary to, not substitutable for, fiscal coordination. The equality-augmented Taylor Rule adds a distributional term to a decision central banks already make, rather than creating a new institutional apparatus.

The model correctly predicts that sufficient fiscal coordination would solve the problem. EIMTS is needed because the required level of fiscal coordination has proven unattainable, and the mechanism design framework explains why. The contribution is not identifying that redistribution reduces inequality — that is trivial — but providing an incentive-compatible institutional architecture that makes coordination self-enforcing.

A diagnostic gap requires explicit acknowledgment. The §3.7 calibration above shows that under OECD-average parameters the bounded Ḡ steady state is approximately 0.33, which is below current within-country Gini levels in most rich nations rather than above them; under this calibration the architecture's role is to reduce Ḡ further toward the politically targeted level rather than to prevent unbounded inequality drift. The diagnosis that actually bites at the level of urgency the architecture is built to address is the global between-country calibration (Ḡ ≈ 0.75–0.85), not the OECD within-country one. But between-country inequality is governed primarily by the growth-and-convergence dynamics across nations — the rate at which low-income countries close the GDP-per-capita gap with high-income countries, which is itself a function of structural-transformation pathways, demographic transitions, terms-of-trade movements, and political-institutional conditions — rather than by any single central bank's reaction function. An augmented national Taylor Rule is structurally a poor match for a between-country target: the rule operates on a domestic policy rate that affects domestic distribution through the channels of §§5.1–5.2, but its effect on the cross-country gap is mediated through trade and capital-flow channels that the national rule does not directly govern.

The architecture's flagship national instrument — the equality-augmented Taylor Rule — therefore addresses the within-country inequality whose magnitude is least pressing under the §3.7 calibration, and is least equipped to move the between-country inequality whose magnitude is most pressing. The honest framing of this is that the §7 transition path's multilateral coordination architecture (Bretton Woods 2.0, the equality clearinghouse, the OECD-equality-tax analogue, the federation-level coordination layer) is where the between-country diagnosis would actually be addressed; the national rule's role is to ensure individual nations are not actively worsening the global picture while the multilateral architecture is constructed. The honest statement of the contribution is that EIMTS at the national-rule level is a containment instrument for within-country drift, and the genuine between-country contribution is the institutional architecture that the rule's adoption enables rather than the rule's mechanism itself. The Conclusion returns to this acknowledgment.

3.8 Corollaries on the EIMTS Constraint

Inequality is bounded above by Ḡ rather than diverging unboundedly; the EIMTS constraint operates by reducing Ḡ below politically tolerable levels, not by preventing unbounded divergence in finite time. The corollaries follow.

Corollary 3.2' (Equality Requires Active Constraint to Lower Ḡ). The asymptotic Gini Ḡ under any architecture is determined by the balance of inequality-increasing forces (Piketty channel, monetary channels, multiplicative variance) and counterforces (taxation, inheritance dilution, entrepreneurial disruption). Reducing Ḡ below the level achievable by uncoordinated national fiscal policy requires an additional structural counterforce that operates through the monetary and trade architecture itself. Specifically: (i) the policy rule must respond to distribution; (ii) seigniorage allocation must redistribute toward those with high Cantillon distance; (iii) trade rules must respond to bilateral equality asymmetry; (iv) capital returns must be constrained where they generate persistent r − g amplification.

Corollary 3.3' (The EIMTS Constraint is Mathematically Sufficient, Not Necessary). The constraint A^ε(x) ≤ A* (the Atkinson-formulation of the equality constraint that will appear in §5.6, recoverable as a Gini-target G* via the Bridging Proposition) is one mathematical formulation of the counterforce required to reduce Ḡ to the politically targeted level G*. It is sufficient to do so under reasonable parameter choices, given the formal equilibrium-existence result of Theorem 5.4'. It is not necessary: alternative architectures, including aggressive multilateral fiscal coordination at scales not yet achieved historically, could in principle achieve the same Ḡ reduction. The case for EIMTS over those alternatives rests on §3.7's coordination-problem argument and on the mechanism-design self-enforcement properties developed in Chapter 4.

This reading is modest and defensible. The EIMTS constraint reduces a bounded inequality steady state to a politically tolerable level; it does not prevent unbounded inequality growth, because no such growth is occurring. The argument is necessary to reach the level of equality the architecture targets, not necessary to prevent catastrophe. That weaker reading is, in fact, sharper.

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§ IVThe Mechanism Design

The two preceding chapters made an architectural case: that the absence of any distributional term in the policy rule (1.1') and in the gravity model (2.4) is a design feature with measurable consequences, and that the consequences cannot be reformed by adjusting parameters within the existing rule because the rule's action space does not include the relevant instrument. Adding such an instrument is straightforward in principle — write γ(G_t − G*) into the Taylor Rule, write E_ij^ζ into the gravity equation, and the architectural problem is resolved at the level of mathematical specification. The genuine difficulty is not specifying the augmented rule. It is showing that any nation-state would have an incentive to participate in an architecture that constrains its own policy choices in this way.

This is the social-coordination problem that mechanism design addresses. The problem has a distinct structure. Nations are sovereign: no global authority can compel participation, and participation cannot be sustained by external enforcement. Nations have private information: each knows its own internal Gini, its own r − g amplification, its own political constraints on redistribution far better than any external observer does. And the social objective — reducing global inequality below a politically tolerable level — depends on aggregate behavior across nations, so unilateral compliance by any single nation is largely ineffective. The architecture must therefore satisfy four properties simultaneously: incentive compatibility (truthful reporting of inequality metrics must be each nation's best strategy), individual rationality (every nation must prefer participation to autarky), budget balance (the system must not require external subsidy), and a fairness property (the allocation must respect some notion of equitable burden-sharing across coalition members). The combination is well-known to be difficult. Green and Laffont (1977) proved that for general type spaces, no mechanism can simultaneously achieve dominant-strategy incentive compatibility (DSIC), ex-post efficiency, and budget balance. The architecture therefore uses the weaker but achievable Bayesian incentive compatibility (BIC) as the default, reaching DSIC only conditionally under a single-crossing condition. The construction is the d'Aspremont and Gérard-Varet (1979, hereafter AGV) expected-externality mechanism, which is to BIC + BB what VCG is to DSIC + efficiency.

4.1 The EIMTS Mechanism: Formal Specification

A mechanism is a formal system for aggregating private information into collective outcomes. We define the EIMTS mechanism as a tuple M = (N, Θ, A, f, t) where N = {1, …, n} is the set of participating nations, Θ = ×_i Θ_i is the product type space, A is the outcome space, f : Θ → A is the allocation rule mapping reported types to outcomes, and t : Θ → ℝ^n is the transfer rule. Each nation's type θ_i = (G_i, HDI_i, Y_i, T_i, r_i^d) encodes its inequality state (the residual Gini conditional on global factors — see §4.4 below), Human Development Index, GDP, trade preferences, and domestic interest-rate preference. Each nation's outcome a_i = f(θ̂) = (r_i*, q_i, x_i) specifies an equality-indexed interest rate, bilateral trade terms, and non-monetary transfers. Each nation receives a transfer t_i(θ̂) ∈ ℝ from a central clearing pool. The mechanism operates through four steps: nations report their types θ̂_i to the coordinator (an institution such as a reformed IMF; we develop the institutional question in Chapter 7); the allocation rule f computes the optimal interest-rate and trade-allocation profile; the transfer rule t calculates AGV transfers; outcomes are implemented and nations receive utility from allocations and transfers.

4.2 The Revelation Principle

The revelation principle (Myerson, 1981) is foundational. For any Bayesian Nash equilibrium of any indirect mechanism, there exists a direct mechanism in which truthful reporting is a Bayesian Nash equilibrium, with identical equilibrium outcomes. The implication is that we lose no generality by restricting attention to direct mechanisms where the strategy space is the type space and where truthful reporting is the target equilibrium. The proof is constructive: if σ*(θ) is the equilibrium strategy in an indirect mechanism with allocation rule f and transfer rule t, define the direct mechanism f'(θ) = f(σ*(θ)), t'(θ) = t(σ*(θ)). Any deviation in the direct mechanism corresponds to a deviation in the indirect mechanism that was not profitable; therefore truth-telling is optimal in the direct mechanism. The remaining work is to construct the specific direct mechanism that achieves the four properties listed above.

4.3 Individual Rationality

A mechanism satisfies interim individual rationality if for all nations i and all types θ_i, the expected utility from truthful participation at least equals expected utility from autarky:

E[v_i(f(θ_i, θ_{−i})) + t_i(θ_i, θ_{−i})] ≥ u_i^autarky(θ_i) (4.8)

We show this holds under the EIMTS mechanism with AGV transfers (defined below). The argument has three components. Standard trade theory proves that moving from autarky to multilateral trade generates aggregate gains: under EIMTS, nations receive v_i^EIMTS = v_i(comparative advantage) + v_i(monetary stability) + t_i. Comparative-advantage gains exceed bilateral arrangements because broader coalitions allow finer matching. Monetary-stability gains are strictly positive for developing economies through reduced exchange-rate volatility and lender-of-last-resort access. The transfer t_i is positive in expectation for nations below the global median residual Gini (after conditioning on global factors); for wealthier nations comparative-advantage gains alone suffice to offset the transfer contribution. Under reasonable parameter calibration, all nations prefer EIMTS to autarky. The full calibration argument is in Appendix A.

4.4 Budget Balance via AGV Transfers

We can now state and prove the central result of this chapter. The transfer rule that achieves budget balance simultaneously with Bayesian incentive compatibility is the AGV (d'Aspremont & Gérard-Varet, 1979) expected-externality mechanism rather than the Vickrey-Clarke-Groves (VCG) mechanism. VCG (Vickrey, 1961; Clarke, 1971; Groves, 1973) achieves DSIC but generically violates budget balance — this is the content of Green and Laffont (1977). AGV achieves BIC + BB simultaneously, which is the combination EIMTS requires.

Define the AGV transfer for nation i as

t_i(θ) = E_{θ_{−i}}[Σ_{j≠i} v_j(f(θ_i, θ_{−i}))] − (1/(n−1)) Σ_{j≠i} E_{θ_{−j}}[Σ_{k≠j} v_k(f(θ_j, θ_{−j}))] (4.AGV)

The first term is nation i's expected positive externality on others; the second term is a constant (from i's perspective) chosen to ensure budget balance. Summing over i, both terms equal (n−1) · E[Σ_j v_j(f(θ))], so E[Σ_i t_i(θ)] = 0: ex-ante budget balance holds.

Ex-post budget balance — Σ t_i(θ) = 0 for every realized type profile θ, not just in expectation — is obtained through the strengthened AGV construction of Krishna and Maenner (2001), which requires (a) independent types and (b) an onto allocation rule. Condition (b) is satisfied by construction: the EIMTS allocation rule maps type profiles to interest-rate adjustments and transfer schedules covering the full outcome space. Condition (a) is the substantive one. National Gini coefficients are not independent: they are correlated through synchronized monetary policy cycles, commodity-price fluctuations, and global recessions. We resolve this by defining types as residual Gini conditional on observable global state. Let Z_t = (g_t^W, p_t^W, i_t^W) denote the global growth rate, commodity-price index, and reference interest rate at time t. Define nation i's type as θ_i = G_i − E[G_i | Z_t], the Gini residual after conditioning on global factors. Under this definition, θ_i captures the idiosyncratic component of each nation's inequality — driven by domestic policy choices, institutional quality, demographic structure, and country-specific shocks. Conditional independence of residuals is standard in panel econometrics under the common-factor methodology developed by Pesaran (2006); it is testable and empirically supportable for the EIMTS application.

If the conditioning set Z_t is misspecified (omitting relevant common factors), residual correlation persists; ex-post BB fails but ex-ante BB still holds. We therefore present the result in two tiers: Theorem 4.7' guarantees ex-ante BB unconditionally, and upgrades to ex-post BB only under the maintained conditional-independence assumption. This is the empirically honest framing — ex-ante BB is the robust result, ex-post BB is available under a testable auxiliary hypothesis.

Fragility caveat on ex-post BB. The Krishna and Maenner (2001) ex-post BB construction requires independent types. The independence assumption we manufacture by conditioning on the global state vector Z_t may be endogenously violated once the architecture is operating: the gravity equality term E_ij in §5.3 couples national distributions further through the trade-volume channel (one nation's bilateral equality is constructed from comparisons with another's, so the residuals after conditioning on Z_t inherit cross-sectional correlation through the architecture's own operation), and the AGV transfers themselves redistribute across nations in a way that mechanically links their realized welfare. The conditional-independence assumption is therefore not merely empirically uncertain — it is fragile by construction. We therefore treat ex-ante BB (which holds unconditionally) as the robust result, and ex-post BB as a fragile-by-construction conditional refinement that an architecture-aware reading should not rely on as a load-bearing guarantee. The architecture remains incentive-compatible at the ex-ante level under the BIC + ex-ante BB combination Theorem 4.7' commits to; the ex-post strengthening should be read as a conditional bonus, not a load-bearing property.

4.5 Bayesian Incentive Compatibility

Under AGV transfers, nation i's expected payoff from reporting θ̂_i is

U_i(θ̂_i, θ_i) = E_{θ_{−i}}[v_i(f(θ̂_i, θ_{−i})) + Σ_{j≠i} v_j(f(θ̂_i, θ_{−i}))] + constant

The bracketed term is the expected total social welfare when i reports θ̂_i. Since f is constructed to maximize total welfare at the truth, U_i is maximized at θ̂_i = θ_i. Therefore truthful reporting is a Bayesian Nash equilibrium. The argument extends the classic Groves construction to the BIC setting: by paying each nation its expected externality on others, the mechanism aligns each nation's reporting incentive with the planner's welfare objective.

Common-prior caveat. Bayesian incentive compatibility presupposes a common prior over the type distribution. Across sovereign nations with strategic incentives over rival beliefs — and with documented incentives in international macroeconomic-statistics reporting to manage how other actors perceive one's economic position — common knowledge of the type distribution is the strongest single assumption the architecture rests on. BIC equilibria are not robust to its failure: under heterogeneous priors, reporting incentives generically diverge from truth-telling, and the mechanism's incentive-compatibility guarantee fails. We flag this explicitly: the incentive guarantees of Theorem 4.7' are only as strong as the common-prior assumption. The IEMA infrastructure of §7.2 is designed in part to support the common-prior assumption by publishing harmonized methodology that all participating nations endorse and operate under; the success of that infrastructure-building is a precondition for the BIC guarantees to bind in practice, not a downstream administrative convenience.

Operational tension between the incentive-compatibility region and the consequential operating region. We surface one further structural tension between the architecture's incentive-compatibility region and its consequential operating region. Assumption 4.A bounds the quasi-linear approximation underpinning AGV transfers to |t_i| ≤ 5% of GDP. But the magnitudes required to move the between-country diagnosis surfaced in §3.7 (Ḡ ≈ 0.75–0.85) and to support the constrained-optimal allocations of Theorem 5.6' (Existence-but-not-Operationality) are 1–2 orders of magnitude above current ~1% of GDP cross-border flows — well outside the 5% bound. The mechanism's incentive guarantees therefore hold in the operating region where transfers are too small to be consequential, and degrade exactly as transfers scale to where they would matter. Resolving this tension — whether by tightening Assumption 4.A's defensible range with empirical estimation of preference quasi-linearity over larger transfers, or by accepting that the incentive-compatible and consequential operating regions may not overlap — is a question this paper poses but does not answer; it is part of what the §7 multilateral coordination architecture would need to settle in its own institutional design.

4.6 Fairness via Shapley Value (Caveated)

The EIMTS transfer rule approximates the Shapley value (Shapley, 1953) — the marginal-contribution-averaged-across-coalitions concept that is the canonical formalization of fair division — with an important caveat. The Shapley value provides a fair division of the cooperation surplus only under symmetric valuation of transfers across coalition members. AGV transfers depend on expected externalities, which are weighted by nations' marginal valuations of transfers. When marginal valuations differ substantially across nations — for example, when the coalition includes both the United States ($25T GDP) and Rwanda ($13B GDP), where a $1B transfer is a rounding error for one and a major budgetary event for the other — the symmetry assumption fails. The Shapley value remains well-defined as a mathematical object but its normative interpretation as "fair division" is weakened.

The repair is to restrict the Shapley-value approximation to size-homogeneous coalitions, where the ratio of largest to smallest GDP is bounded (empirically, a ratio of 10:1 gives adequate symmetry). For the full heterogeneous coalition the Shapley value is replaced by the nucleolus (Schmeidler, 1969) as the fair-division concept, which does not require symmetric valuations. The practical implication is institutional: EIMTS is best designed as a federation of size-homogeneous coalitions (advanced economies, large developing economies, small open economies) linked by a top-level coordinating layer, rather than as a single flat coalition of all participating nations. Chapter 7's transition path develops this institutional structure.

4.7 Social Choice and Arrow

Arrow's (1951) impossibility theorem proves that no social-choice rule can simultaneously satisfy unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives when preferences are purely ordinal. EIMTS escapes the impossibility because it operates on cardinal, interpersonally comparable capabilities — Gini, HDI, GDP — rather than ordinal preferences. The escape route is Sen's (1999) capabilities approach, which substitutes capability comparison for preference aggregation. Because the EIMTS allocation rule depends on observable cardinal measures that are interpersonally comparable in principle (the residual Gini, conditional on Z_t, is computed identically for every nation), Arrow's framework simply does not apply. This is the canonical justification for inequality-targeting in mechanism design and is uncontroversial in the welfare-economics literature.

4.8 Theorem 4.7': EIMTS Mechanism Existence

We synthesize the preceding analysis. DSIC + ex-post BB + ex-ante efficiency cannot hold jointly for general type spaces (Green-Laffont, 1977); the theorem therefore commits to BIC as the default and gives DSIC as a conditional strengthening.

Theorem 4.7' (EIMTS Mechanism Existence — Revised). There exists a direct mechanism M = (N, Θ, A, f*, t*) implementing the Equality-Indexed Monetary-Trade System such that:

(i) Bayesian Incentive Compatibility (BIC): truthful reporting is a Bayesian Nash equilibrium;

(ii) Interim Individual Rationality (IIR): all nations prefer participation to autarky in expectation;

(iii) Ex-ante Budget Balance (BB): E[Σ t*_i(θ)] = 0, with ex-post BB obtaining as a fragile-by-construction refinement under the conditional-independence assumption documented in §4.4;

(iv) Ex-ante Efficiency: the allocation f* maximizes expected social welfare subject to (i)–(iii).

A DSIC strengthening via the Myerson (1981) payment identity under smooth single-crossing is not available because the EIMTS type space defined in §4.1 is five-dimensional — θ_i = (G_i, HDI_i, Y_i, T_i, r_i^d) — and the Myerson one-dimensional payment identity does not apply to multi-dimensional types without first projecting θ_i onto a scalar index and justifying the projection economically. A single-crossing argument via Galor-Zeira credit-constraint channels establishes monotonicity along the G_i dimension but says nothing about the joint single-crossing structure required for multi-dimensional DSIC.¹ Theorem 4.7' therefore stops at BIC + IIR + ex-ante BB + ex-ante efficiency and does not claim DSIC. The architecture remains incentive-compatible at the Bayesian-Nash level under the AGV construction; the dominant-strategy upgrade requires structural work this paper does not do.

The proof sketch is the engine. The AGV transfer formula above (eq. 4.AGV) is the mechanism's load-bearing construction: by paying each nation its expected externality on others, the system simultaneously aligns reporting incentives with social welfare maximization (BIC) and ensures that aggregate transfers balance in expectation (ex-ante BB). Ex-post BB obtains as a fragile refinement (per §4.4) only under the conditional-independence assumption that the architecture itself partly undermines through the gravity-channel coupling and the AGV-transfer coupling. IIR follows from the gains-from-trade calculation under expected transfers (§4.3). Efficiency follows from f*'s construction as the maximizer of expected social welfare, with the revelation principle ensuring no other mechanism can do better while satisfying BIC + BB. The full proof bodies appear in Appendix A.4.

Remark on Quasi-Linearity. The AGV and VCG mechanisms both assume quasi-linear preferences: u_i = v_i(a) + t_i. For sovereign nations with complex political objectives this is a strong assumption. We make it operative through Assumption 4.A (Approximate Quasi-Linearity): national utility functions satisfy u_i(a, t_i) = v_i(a) + w_i(t_i) where w_i is concave with w_i'(0) = 1. For transfers |t_i| ≤ 0.05 × GDP_i, the departure from linearity satisfies |w_i(t) − t| < 0.01 × |t|. This is empirically calibratable: the assumption states that marginal valuation of transfers is approximately constant for transfers up to 5% of GDP, which is well within the range of existing international transfers (EU structural funds are 1–2% of recipient GDP; Marshall Plan transfers peaked at roughly 2.5% of recipient GDP). The standard justification for quasi-linearity in public-goods mechanism design (Milgrom, 2004) applies.

4.9 Discussion

The honest reading of Theorem 4.7' is that EIMTS is implementable under conditions that are restrictive but defensible: Bayesian Nash equilibrium is the solution concept, AGV is the transfer rule, types are residual Gini conditional on global state, and the size-homogeneity restriction on the Shapley-value interpretation pushes the architecture toward a federated rather than flat coalition structure. The mechanism is not self-implementing in the strong sense — it requires a coordinating institution to compute and disburse AGV transfers, and it requires nations to maintain the conditioning-set Z_t with shared methodology. But it is not externally enforced either: once nations enter, no nation has incentive to misreport, and no nation has incentive to defect for any single-period gain that does not exceed the discounted value of continued participation. This is qualitatively different from the situation of voluntary fiscal coordination that we contrasted EIMTS with in §3.7.

The remaining question is whether the architecture, given its incentive properties, would actually be adopted. That is the question of Chapter 7. Before reaching it, we have to specify the architecture itself in unified form — the central work of Chapter 5.

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§ VThe Unified EIMTS Model

The architectural case made in Chapters 1 and 2 and the mechanism-design feasibility result of Chapter 4 between them establish that an equality-indexed monetary-trade system can exist as a formal object. This chapter specifies what such a system looks like in unified form. Four instruments — the equality-augmented Taylor Rule (§5.1), the equality-indexed money supply rule (§5.2), the equality-adjusted gravity model (§5.3), and the fair-value exchange-rate monitoring band (§5.4) — sit inside a single constrained general-equilibrium problem (§5.5), which has an equilibrium under conditions stated in Theorem 5.4' (§5.6). Welfare theorems establish that the equilibrium is constrained Pareto optimal, that the second-welfare-theorem analogue holds as a non-operational existence result (Theorem 5.6'), and that the constrained equilibrium delivers a conditional aggregate (Kaldor-Hicks) welfare improvement over the unconstrained competitive equilibrium when the inequality-externality magnitude clears a contested lower bound (Theorem 5.7''; see §5.7). Uniqueness and comparative statics close the chapter (§5.8). The chapter is the largest in the paper because it is the load-bearing specification: every empirical claim in Chapter 6 and every transition-path commitment in Chapter 7 traces back to one of the formal objects defined here.

5.1 The Equality-Augmented Taylor Rule

The standard Taylor Rule responds to inflation and output gaps. The simplest expression of EIMTS at the monetary level is to add a third term that responds to distribution. The mechanism is direct: when actual inequality exceeds the target, the central bank reduces the policy rate, stimulating employment and wage growth in the parts of the distribution most exposed to monetary policy's labor-market channel; when inequality falls below target, the central bank can raise rates without conflicting with its equality objective. The augmentation is conceptually small — one term added to the right-hand side of a familiar equation — but the architectural implication is large: it enlarges the rule's action space to include the distributional response Theorem 1.1' showed was absent.

Formally, the equality-augmented Taylor Rule is

i_t = r* + α(π_t − π*) + β(y_t − y*) + γ(G_t − G*) (5.1)

where the additional term γ(G_t − G*) is the equality response, G_t is the current disposable-income Gini coefficient (measured per the standardized methodology of WID — Alvaredo et al., 2018; Chancel et al., 2022 — and the Luxembourg Income Study microdata harmonization; technically the formally binding constraint is A^ε(x) ≤ A* via the Bridging Proposition, with x the disposable-income distribution and we retain Gini-language here as the reporting metric — see §5.6), G* is the target disposable-income Gini coefficient (typically 0.25–0.35 in advanced-economy calibrations), and γ is the equality response coefficient. The welfare-optimal γ is derived by minimizing the central bank's loss function L = E_t Σ δ_L^k [π_{t+k}² + λ_y(y_{t+k} − y*)² + μ_G(G_{t+k} − G*)²] with respect to γ, where μ_G is the equality weight in the loss function (see Appendix A.1 for the notation glossary), yielding γ* = μ_G / (κ_1 + κ_2 γ_1) where κ_1 and κ_2 are the semi-elasticities of inequality with respect to interest rates and inflation respectively. Empirical estimates put γ* in the range 0.15–0.25 for modern economies.

The system's three-dimensional dynamics — inflation, output, equality gap — exhibit saddle-path stability whenever γ < γ̄ where γ̄ is determined by the system Jacobian. Calibration under standard parameters yields γ̄ ≈ 1.70, comfortably above any politically plausible value of γ; saddle-path stability is therefore not a binding constraint in calibrated implementations. (The full stability derivation is in Appendix A.6.) The welfare comparison is straightforward: when inequality increases exogenously, the standard rule cannot respond, leaving inflation and output to adjust through general-equilibrium channels; the augmented rule directly addresses the source of welfare loss, reducing required volatility along the other two dimensions. The augmented rule strictly dominates the standard rule when inequality shocks are present, and equals it when they are absent.

What the augmentation does — beyond the formal welfare improvement — is route the Cantillon channel of §1.2 through a feedback loop. When inequality rises, the augmented rule lowers i, which stimulates employment (reducing inequality through the labor-market channel) and raises asset prices (which by itself would raise inequality through the Cantillon channel). The empirical question is which channel dominates; the HANK literature (Auclert, 2019; Bayer, Born, & Luetticke, 2024) suggests the labor-market channel dominates the disposable-income measure at horizons longer than a quarter or two, which is the policy-relevant horizon. The augmentation is therefore not a guarantee of inequality reduction at every moment; it is a feedback loop that bends the long-run trajectory of the disposable-income Gini toward G*.

The wealth-Gini side effect

The commitment to disposable income Gini as the target object requires an explicit acknowledgment of what the architecture does not target. The rate-cut channel that improves disposable income Gini through labor-market stimulation simultaneously raises asset prices, and asset-holding concentration in the top decile (Federal Reserve, 2022) means that the wealth distribution moves in the opposite direction. Auclert (2019), Andersen, Johannesen, Jørgensen, and Peydró (2023), and Acharya, Challe, and Dogra (2023) document the mechanism quantitatively: the bond-revaluation channel, the equity-revaluation channel, and the housing-revaluation channel each transmit accommodative monetary policy disproportionately to the top-decile balance sheet. The disposable-income Gini and the wealth Gini therefore move in opposite directions during an expansionary cycle, and the architecture as specified — operating entirely through the policy-rate channel — cannot resolve the asymmetry on its own.

The honest framing of this is that EIMTS targets the flow distribution (disposable income, captured by WID and LIS at the consistent definition the architecture commits to) and accepts that the stock distribution (wealth) requires a complementary architecture. The complementary instrument set is macroprudential rather than monetary: loan-to-value (LTV) and debt-service-to-income (DSTI) caps on the real-estate channel, tightened differentially when the wealth Gini rises faster than the income Gini falls; differential capital surcharges on bank lending against asset-collateralized loans relative to lending against productive-investment collateral; transaction-tax adjustments that distinguish primary-residence transactions from investment-property transactions. These tools sit inside the macroprudential mandate of central banks and financial-stability authorities (Basel III/IV framework, the IMF's macroprudential toolkit guidance) rather than inside the monetary mandate, and their natural institutional home is the macroprudential committee — typically a separate body from the monetary policy committee, though resident at the same central bank — coordinated with the monetary policy through the joint financial-stability framework.

The architecture's commitment, stated explicitly: EIMTS as specified addresses disposable income Gini directly and acts as a stabilizing pressure on wealth Gini only through the labor-channel (rising wages eventually translate into rising wage-saver wealth at the bottom of the distribution) and through the macroprudential complementarities described above; a paper specifying the joint architecture is open work. The §7 transition path stages the macroprudential coordination alongside Phase II's monetary-rule adoption: the IEMA (§7.2) would extend its harmonized-reporting mandate to wealth-distribution metrics as a Phase I deliverable, and Phase II's coalition-level commitment would include the macroprudential complementarity from the outset rather than as a Phase III add-on. The reason for staging this jointly with Phase II rather than deferring to Phase III is that the wealth-channel disequilibrium produced by §5.1's rate-cut response would otherwise accumulate over Phase II and become a political liability for the architecture in Phase III adoption.

Tinbergen instrument-target accounting

The architectural design also requires explicit accounting of which instrument addresses which target. The Tinbergen (1952) assignment rule states that a policy authority needs as many instruments as it has targets; this is sometimes read as ruling out an equality-augmented Taylor Rule on the grounds that it would attempt to hit three targets (inflation, output, equality) with a single instrument (the policy rate). The honest framing is that the EIMTS architecture has more than one instrument and treats the targets as constrained rather than as multiplicatively independent. The table below reflects two refinements: the i / M independence tension under the §1.4 endogenous-money commitment (worked through in §5.2.x), which collapses the separate i and M entries into one monetary lever with price (i) and quantity (M_base) sides; and an explicit macroprudential row corresponding to the wealth-Gini target the §5.1 side-effect subsection above introduces.

| Instrument | Primary Target | Secondary Target | Tertiary Target |

|---|---|---|---|

| Policy rate i (price side of monetary lever; eq. 5.1) | π (inflation) | y (output gap) | G (disposable-income Gini) |

| Monetary base M_base (quantity side of monetary lever; eq. 5.9 — operative at ZLB or as institutional signal) | G (disposable-income Gini, via Ψ(E_t)) | Signaling on equality dimension | — |

| Gravity ζ-augmentation (trade; eq. 5.15) | Trade-equality (E_ij) | Bilateral surplus distribution | — |

| Macroprudential committee (LTV / DSTI / capital surcharges — outside monetary core) | G_w (wealth Gini) | Financial stability | — |

| FX monitoring band (§5.4 — non-binding) | Bilateral fair-value reporting | — | — |

The accounting frames G as a third target in a constrained optimization rather than as a fourth instrument requiring its own dedicated control. The constrained-optimization framing makes explicit that the architecture accepts trade-offs along the (π, y, G) Pareto frontier rather than claiming to hit all three simultaneously, and locates the equality response inside the same Tinbergen accounting that already governs the dual mandate. Two clarifications follow from the table. First, the policy rate i and the monetary base M_base are listed as the price and quantity sides of a single monetary lever rather than as two independent instruments; the i/M independence tension that the §1.4 endogenous-money commitment surfaces is worked through in §5.2.x below, where the implication for the Tinbergen accounting (two operational levers — monetary and trade) is stated explicitly. Second, the macroprudential row sits outside the formal monetary-trade architecture (it is staged in the §7 transition phases as a complementary institutional layer); it is included in the accounting to register the wealth-Gini target the §5.1 side-effect subsection introduces and to acknowledge that the income-Gini target the rate rule pursues and the wealth-Gini target the macroprudential committee pursues are coordinated across institutional boundaries rather than within a single rule. The architecture is Tinbergen-compatible by construction.

5.2 The Equality-Indexed Money Supply

The Taylor Rule operates on the price of money. The money-supply rule operates on the quantity. EIMTS modifies the latter to respond endogenously to a broader composite equality index, implementing a kind of Gesellian demurrage incentive through the monetary channel (Gesell, 1916; Champ, 2008): when equality worsens, the implicit holding cost of money rises by contraction of the money supply, incentivizing productive deployment of capital and labor; when equality improves, the supply expands, accommodating the more-equal distribution's higher marginal velocity.

Formally, the money supply at time t is M_t(eq) = M_t(base) · Ψ(E_t) where M_t(base) is the exogenous monetary base and Ψ(E_t) = exp(λ_M · (E_t − E*)) is the equality adjustment factor, with λ_M the money-supply equality sensitivity parameter (typically 0.5–1.5; see Appendix A.1 for the notation glossary) and E* the target composite equality index. The composite equality index combines distributional, capability, and environmental dimensions: E_t = w_1(1 − G_t) + w_2 · HDI_t + w_3 · GPI_t / GDP_t, with default weights w_1 = 0.50 (inverse disposable-income Gini), w_2 = 0.30 (Human Development Index), and w_3 = 0.20 (Genuine Progress Indicator relative to GDP). The composite captures three substantive dimensions of equality — income distribution, human capability, sustainable progress — and the weights reflect their relative importance in EIMTS social-welfare prioritization (which is itself a policy parameter, not a technocratic given).

Caveat on GPI as composite component. The Genuine Progress Indicator is a contested non-standard construct with multiple competing implementations (Talberth, Cobb, & Slattery, 2007; Kubiszewski et al., 2013) and substantial cross-national methodological disagreement on what counts as "defensive expenditure," "ecological cost," or "non-market production." Including GPI as a binding component of the money-supply rule imports measurement controversy directly into the binding rule. A more conservative alternative implementation would substitute the World Bank's Adjusted Net Savings (ANS) measure (a standardized sustainability indicator with consistent cross-country methodology) or the UN Environment Programme's Inclusive Wealth Index (IWI, which extends national accounts to include natural and human capital depreciation). The default specification retains GPI for continuity, but an implementation-ready version of the architecture should substitute ANS or IWI for the GPI/GDP component to reduce the measurement-dispute surface area. The §5.1 Tinbergen accounting framing treats E_t as a composite-equality target; the specific composition is a policy choice rather than a structural commitment of the architecture.

The money supply expansion property dM_t(eq) / dE_t = λ_M · M_t(base) · Ψ(E_t) > 0 creates a positive feedback loop: improved equality enables monetary expansion that further reduces inequality through lower interest rates and higher employment. The inflation constraint π_t = (dM_t / M_t) − (dY_t / Y_t) + (dV_t / V_t) bounds the speed of Ψ adjustment; for typical calibrations λ_M ≤ 1.5 keeps monetary growth within the bounds set by real growth plus inflation target. The inflation-targeting regime is preserved.

What this enables is a second-order architectural shift: the monetary base is no longer a purely macroeconomic instrument indexed to aggregate prices and output, but a vector instrument indexed to a multidimensional equality measure. The architecture says, in effect, that the central bank's job is to maintain price stability and distributional stability, and that the two are not in fundamental tension at policy-relevant horizons. The HANK literature has begun to make this case in the heterogeneous-agent framework (Kaplan, Moll, & Violante, 2018; Acharya, Challe, & Dogra, 2023; Debortoli & Galí, 2024); EIMTS provides the architecture-level expression of the same insight.

§5.2.x On the i / M Independence Tension

The Tinbergen instrument-target table of §5.1 lists the policy rate i and the monetary base M_base as the price and quantity sides of a single monetary lever rather than as two independent instruments. The reframing is forced by the §1.4 commitment to endogenous money: treating i and M as separately controllable levers is inconsistent with the endogenous-money literature the §1.4 framing endorses.

The substantive tension is that the policy-rate rule (5.1) and the equality-indexed money supply rule (5.9) cannot be enforced simultaneously as two independent instruments. At the rate (5.1) prescribes, the money stock the system delivers will generically differ from M_t(base) · Ψ(E_t), because under endogenous money (BoE 2014; Kelton 2020; the literature §1.4 cites) broad money is demand-determined and is not a central-bank instrument at all. The M in equation (5.9) accordingly refers to the monetary base, which the central bank does control via open-market operations and reserve creation. But the same §1.4 endogenous-money commitment that displaces the multiplier framing also severs the textbook multiplier-driven link from base to broad money. The transmission from base-money adjustment via Ψ(E_t) to the equality target is therefore weak and uncertain in the same way that the §1.4 critique attributes weak transmission to base-money adjustment generally — broad money expansion follows credit demand at margins the central bank does not directly set, and the base-side equality response of (5.9) operates through whatever portion of base expansion is intermediated into credit conditions that pass through to the disposable-income distribution.

The honest accounting is that the two monetary rules (5.1) and (5.9) describe two aspects of a single monetary lever rather than two independent levers. The rate rule (5.1) handles the dual mandate with the equality augmentation as the §5.1 table's tertiary target; the base-money rule (5.9) operates as a quantity-side signal-and-complement that communicates institutional intent on the equality dimension and provides operational space where the rate is constrained by the zero lower bound. The ZLB case where (5.9) becomes operative is in fact the case where the rule is most useful: when the rate cannot move further but the base can be expanded with explicit distributional intent (the equality-indexed analogue of QE — the asset-purchase composition selected to lean against rather than amplify the wealth-channel side effect of §5.1), the quantity-side lever supplies the operational space the price-side lever has run out of.

The Tinbergen accounting therefore reduces to two operational instruments: one monetary (combining i and M_base as price and quantity sides of the same lever) and one trade (ζ via the gravity augmentation of §5.3). The equality objective competes within the monetary lever rather than getting its own; the §5.1 table's "tertiary target" framing of G under the rate rule, and the §5.1 table's "primary target" framing of G under the base rule, reflect the same underlying lever's two aspects rather than two independent assignments. The §5.4 fair-value monitoring band, listed in the table for completeness, is non-binding (per the trilemma honesty of §5.4 below) and does not enter the constrained-optimization program as a control instrument. The macroprudential row of the table likewise sits outside the formal monetary-trade architecture and addresses the wealth-Gini target that the income-Gini-targeting monetary lever does not reach.

5.3 The Equality-Adjusted Gravity Model

International trade responds not only to economic size and distance but also to bilateral equality conditions. The standard Anderson-van Wincoop gravity equation (eq. 2.27) needs a single modification to express the EIMTS architecture at the trade level: an equality term E_ij raised to elasticity ζ (renamed from γ to avoid the Taylor-Rule overloading; see Appendix A.1).

The equality-adjusted gravity equation is

T_ij = A · (Y_i · Y_j / Y_world) · (t_ij / (Π_i · P_j))^(1−σ) · E_ij^ζ (5.15)

where T_ij is bilateral trade flow, A is the productivity parameter, σ is the elasticity of substitution, and E_ij is the bilateral equality index. The exponent (1−σ) is the standard Anderson-van Wincoop (2003) form consistent with equation (2.27). The elasticity ζ is treated here as a free parameter pending estimation, not as a pilot estimate; we use ζ ≈ 0.15 as an illustrative reference value drawn from analogous equality-gradient elasticities in the trade-cost literature, with the understanding that estimating ζ on actual bilateral-trade data is open empirical work. The bilateral equality index combines three dimensions: E_ij = φ(−|G_i − G_j|) · φ(HDI_i + HDI_j) · φ(LS_ij), where the first factor penalizes disparate disposable-income distributions, the second rewards high joint human development, and the third rewards harmonized labor standards. The function φ is a logistic mapping to [0,1].

Honesty caveat on structural interpretation. Equation (5.15) as written is an empirically motivated augmentation of the gravity equation rather than a full structural re-derivation. The Anderson-van Wincoop (2003) multilateral resistance terms Π_i, P_j are endogenous to the bilateral trade-cost matrix; introducing E_ij^(1−σ) into the trade cost (or, equivalently, multiplying the standard gravity prediction by E_ij^ζ) requires re-solving the full general-equilibrium fixed point for (Π_i, P_j) under the augmented cost structure. The augmentation is consistent with the structural skeleton — it preserves the gravity functional form and the multilateral-resistance term-structure — but it does not preserve the closed-form structural interpretation, because the fixed-point system for (Π_i, P_j) is no longer the system Anderson-van Wincoop solved. A full structural integration would proceed by extending the Anderson-van Wincoop fixed-point system to include the E_ij factor in the trade-cost matrix and solving the resulting equilibrium for the joint (Π, P, E) vector. That integration is open work; the paper retains the reduced-form augmentation with this caveat explicit rather than asserting a structural derivation it has not done.

The trade-volume property dT_ij / dE_ij = ζ · T_ij / E_ij > 0 provides material incentive for trading partners to improve distributional outcomes jointly. Countries converging toward bilateral equality gain increased trade volumes and the associated gains from trade. The empirical estimation specification is log-linear: ln(T_ij) = α + β_1 ln(Y_i · Y_j) − β_2 ln(D_ij) + ζ ln(E_ij) + τ_i + τ_j + ε_ij, with country fixed effects τ_i, τ_j. At the illustrative reference value ζ ≈ 0.15, a 10% increase in bilateral equality (through Gini convergence, HDI improvement, or labor-standards harmonization) would increase bilateral trade by approximately 1.5%; the actual elasticity awaits estimation.

What this enables — beyond the trade-volume incentive — is the architectural shift on the trade side analogous to the monetary side. Trade rules now respond to distributional outcomes; the gravity model's action space is enlarged to include the equality response that Theorem 2.4' established was absent. The two architectures (monetary and trade) reinforce each other: nations that improve internal distribution simultaneously gain monetary accommodation (through §5.1's lower γ(G_t − G*) penalty) and trade access (through §5.3's higher E_ij^ζ multiplier). The joint mechanism is the load-bearing claim of the unified model.

5.4 The Fair Exchange-Rate Monitoring Band

The fourth EIMTS instrument reports bilateral exchange rates against a fair-value benchmark derived from equality-adjusted fundamentals, providing a monitoring band that flags large deviations as transparency information rather than binding policy targets. The reframing to a monitoring band with intervention only at extreme edges is the explicit consequence of taking the Mundell-Fleming trilemma seriously, as we develop below.

The fair bilateral exchange rate from currency i to currency j is e_ij = PPP_ij · Φ(HDI_ij, G_ij, XI_ij), where PPP is purchasing-power parity, HDI is the human development ratio, G is the disposable-income Gini ratio, and XI is an environmental-sustainability index. The equality adjustment Φ is constructed so that the fair rate is higher than PPP when the developing economy has lower HDI or higher inequality relative to the developed counterpart — the wedge reflects the developing economy's need for exchange-rate accommodation to support employment-intensive growth and capability development. The monitoring band δ_ij defines the reporting range: e_ij ∈ [e_ij(1 − δ_ij), e*_ij(1 + δ_ij)] with δ_ij set wide enough to be non-binding under ordinary fluctuations. Our default specification sets δ_ij ≈ 0.30 (a thirty-percent band around fair value), narrowing as bilateral equality converges only at the reporting level (the band labels deviations as "within fair-value range" or "outside fair-value range" without triggering automatic intervention). Intervention is contemplated only at the extreme edges (deviations above ±30% from fair value, sustained over a multi-quarter window), and even there the intervention is consultative through the IEMA (§7.2) framework rather than a symmetric-counter-movement mandate.

Trilemma honesty. The reframing is forced by the Mundell-Fleming trilemma. A binding fair-value corridor coexisting with the equality-targeting monetary rule of §5.2 and open capital accounts would violate the trilemma — fixed exchange rates, independent monetary policy, and free capital mobility cannot coexist, as the ERM crisis of 1992 demonstrated canonically and as the literature has reconfirmed in every subsequent episode (the 1997 Asian crisis, the 2010–2015 eurozone periphery crises, the 2018–2022 emerging-market currency runs). A binding corridor with symmetric counter-movement would be incompatible with the open-capital-account assumption underlying the §5.2 monetary rule's independent operation. The architecture as specified therefore makes the corridor non-binding, which preserves the trilemma compliance: independent national monetary policy and open capital accounts continue to operate, with the exchange rate adjusting as a market-clearing variable that the monitoring band observes rather than constrains. The "beggar-thy-neighbor prevention" framing of a binding corridor is set aside because a non-binding monitoring band cannot prevent competitive devaluation by definition; what it does is report deviations transparently, providing the informational base for diplomatic and trade-policy response through other channels (the §5.3 gravity ζ-augmentation, the §7 multilateral coordination architecture).

The path to a binding fair-value corridor — if that is ever the desired institutional outcome — requires either (i) coordinated capital controls across coalition members, treated structurally in §7.6 below rather than as a residual risk; or (ii) coordinated multilateral implementation of monetary policy such that the §5.2 rule operates at the coalition level rather than the national level, which is beyond this paper's scope and would constitute a separate architectural commitment akin to monetary union. Neither option is incorporated into the present architecture; both are flagged as natural extensions in future work.

The architectural ancestor for the monitoring band is the IMF's Article IV surveillance framework and the more recent macroeconomic-imbalances reporting under the EU's six-pack regulation, which report exchange-rate deviations and current-account imbalances against benchmarks without triggering automatic intervention. The ERM is the binding version that the trilemma rules out under open capital accounts; the IMF-Article-IV analogue is the non-binding monitoring counterpart that the architecture commits to.

5.5 The EIMTS General Equilibrium

The four instruments above are not independent. They sit inside a single constrained general-equilibrium problem whose formal specification — the EIMTS economy E = (I, J, (ω_i), u_i, (Ω_j), Y_j, p*, i*, W, A*) — is deferred to Appendix A (Definition A.1, with constraints A.1.1–A.1.7 restating the Atkinson constraint A^ε(x) ≤ A* per the Bridging Proposition of §5.6, and three welfare forms — Rawlsian (eq. A.32), capabilities-based (eq. A.33), and generalized CES (eq. A.34) — listed in Appendix §A.3). The four EIMTS instruments — equations (5.1) for the equality-augmented Taylor Rule, (5.9) for the equality-indexed monetary base rule, (5.15) for the equality-adjusted gravity model, plus the non-binding monitoring band of §5.4 (which surveils but does not constrain the equilibrium program) — together with the equality constraint specified at the constitutional level enter the formal optimization developed as Definition A.1 in Appendix A. The monitoring band of §5.4 is therefore not a constraint in the equilibrium definition (consistent with Definition A.1, which omits any corridor object from A.1.1–A.1.7); it is a reporting-and-publication instrument operating outside the formal constrained-optimization program. The remainder of this chapter takes the equilibrium concept as given and develops its existence (§5.6), welfare properties (§5.7), and uniqueness / comparative statics (§5.8). The formal specification is deferred to Appendix A; the substantive content of the remainder of this chapter is that this equilibrium exists, is welfare-improving over the unconstrained baseline, and is locally unique under the conditions stated below.

5.6 Equilibrium Existence and the Atkinson/Gini Substitution

Theorem 5.4' (equilibrium existence for the EIMTS economy) commits to A^ε(x) ≤ A* as the binding object — the Gini's level sets are not convex in general, whereas the Atkinson sub-level sets are for ε > 1 — and recovers the Gini target via a Bridging Proposition that is exact for log-normal distributions and conservative (containment-based) for general distributions.

Theorem 5.4' (EIMTS Equilibrium Existence). For the EIMTS economy E = (I, J, (ω_i), u_i, (Ω_j), Y_j, p*, i*, W, A*) where the equality constraint is specified as A^ε(x) ≤ A* for inequality aversion parameter ε > 1, an equilibrium (p*, i*, x_1*, …, x_n*) exists.

Bridging Proposition. For any target Gini G*, there exists a corresponding Atkinson threshold A*(ε, G*) such that the Atkinson constraint A^ε(x) ≤ A* is a sufficient condition for G(x) ≤ G†, where G† is a computable function of A* and ε. For log-normal income distributions, both G and A^ε are monotone increasing functions of the log-variance σ², establishing a bijection between them: given G*, solve G* = 2Φ(σ/√2) − 1 for σ², then compute A* = 1 − exp(−εσ²/2). The worked example below uses ε = 2 throughout (a standard inequality-aversion choice corresponding to a moderate-aversion social welfare function).

The proof of Theorem 5.4' uses standard Steps 1–3 (excess demand definition, upper-hemicontinuity via Berge's Maximum Theorem, Walras' Law) and a revised Step 4. Under the Atkinson constraint with ε > 1, the feasible set S = {x ∈ ℝ_+^n : A^ε(x) ≤ A*} is convex. The argument: the equally-distributed-equivalent-income function Y_ede = [Σ_i (x_i)^(1−ε) / n]^(1/(1−ε)) is concave in x for ε > 1 (the inner sum is convex; the outer power 1/(1−ε) ∈ (−1, 0) is concave and decreasing on ℝ_+; the composition of a concave decreasing function with a convex function is concave by standard convex-analysis arguments — Boyd & Vandenberghe, 2004, §3.2.4). Since A^ε(x) = 1 − Y_ede / μ and Y_ede is concave, the sub-level set {x : A^ε(x) ≤ A*} is the super-level set of a concave function and hence convex. Steps 5–6 (Kakutani fixed-point and equilibrium construction) proceed in the standard way with the Atkinson constraint as the binding object throughout. The Bridging Proposition is proved exactly for log-normal distributions via the shared σ² and by Lorenz-domination containment for general distributions, drawing on Atkinson (1970), Thistle (1989), and Lambert (2001). Full proof bodies are in Appendix A.4.

What this enables is that the rest of the model can keep talking about a Gini target G* = 0.30 as a reporting metric while the formally binding equilibrium constraint is the convex A^ε(x) ≤ A*. The substitution is honest about what the math actually permits: the equilibrium-existence proof requires convexity, the Gini does not in general have convex level sets, and the Atkinson with ε > 1 does. The Bridging Proposition lets policy audiences continue to think in Gini terms.

5.7 Welfare Theorems

Three welfare theorems characterize the EIMTS equilibrium. Their statements appear here; the proofs are in Appendix A.5.

Theorem 5.5 (Modified First Welfare Theorem). An EIMTS equilibrium allocation is constrained Pareto optimal subject to the equality constraint A^ε(x) ≤ A*. No alternative allocation can make any household strictly better off without either making another household worse off or violating the equality constraint.

Theorem 5.6' (Existence-but-not-Operationality of Constrained-Optimal Allocations). For any constrained-optimal allocation x* satisfying the EIMTS welfare constraints — including A^ε(x*) ≤ A* and any auxiliary capability or sustainability constraints invoked by the chosen W in §A.3 — there exist initial-endowment redistributions Δω that would support x* as a competitive equilibrium of the constrained economy. We state and prove the existence; we do not claim operationality. Under the seigniorage capacity analyzed in §1.4 (commercial-bank-credit creation operating endogenously, with the central bank's direct seigniorage capped by the inflation constraint) and under the redistribution-magnitude calibration of §3.7 (existing cross-border fiscal transfers at approximately 1% of recipient GDP versus the architecture's required magnitudes one-to-two orders of magnitude above that), the required Δω is not feasible from monetary financing alone. Theorem 5.6' is therefore a non-operational existence result: the second-welfare-theorem structure holds for the constrained-equilibrium model, but its policy operationalization requires fiscal complementarity outside the architecture's monetary core. The theorem makes this explicit: seigniorage financing alone does not bridge the gap to the required Δω.

Theorem 5.7'' (Conditional Kaldor-Hicks / Aggregate Welfare Improvement). Conditional on the inequality externality function E(·) satisfying E(G_CE) − E(G*) ≥ Ē, where Ē is the lower-bound magnitude required to clear the aggregate compensation requirement under the constrained allocation, the EIMTS equilibrium produces an aggregate (Kaldor-Hicks) welfare improvement over the unconstrained competitive equilibrium baseline. The improvement does not constitute a Pareto improvement at the household level unless explicit compensating transfers are paid to consumption losers — and per Theorem 5.6' (Existence-but-not-Operationality) those transfers are not operational under current seigniorage and fiscal constraints. The theorem therefore supplies a potential-Pareto / aggregate-welfare argument, not an actual-Pareto ordering at the household level. The lower bound Ē is the contested empirical magnitude of the architecture's welfare claim; Theorem 5.7'' operates as a sufficient condition for aggregate Kaldor-Hicks improvement, not as an unconditional welfare ordering.

The intuition of Theorem 5.7'' is the welfare-economic case for EIMTS in compact form. In the unconstrained competitive equilibrium, agents do not internalize the externality cost of inequality — psychic costs of relative deprivation, reduced trust and social cohesion, lower aggregate productivity, political-instability risk. These externalities reduce aggregate utility. The EIMTS constraint internalizes them by restricting allocations to more equal distributions; the efficiency cost shows up as binding consumption reductions for households at the top of the distribution, and the welfare comparison turns on whether the externality reduction E(G_CE) − E(G*) is large enough to compensate those losses in aggregate terms. When it is, the aggregate (Kaldor-Hicks / potential-Pareto) improvement is delivered; whether it becomes an actual Pareto improvement at the household level depends on whether compensating transfers are paid to the losers, and Theorem 5.6' tells us those transfers are not operational under current seigniorage and fiscal constraints. The theorem supplies the aggregate-welfare argument; the actual-Pareto ordering would require the operationally-infeasible compensations.

E(·) is posited rather than estimated. The sign dE/dG > 0 is well-documented, but the sign does not bound the magnitude of E(G_CE) − E(G*), and the magnitude must clear the threshold Ē for the welfare-improvement claim to be defensible. Even when Ē is cleared, what is delivered is Kaldor-Hicks / aggregate-welfare improvement rather than actual-Pareto improvement at the household level, because the compensating transfers Theorem 5.6' specifies are not operational. The literature on the inequality externality (Wilkinson & Pickett, 2009 on health and social cohesion; Stiglitz, 2012 on the productivity and political-stability channels; Acemoglu & Robinson, 2019 on institutional erosion) supplies suggestive evidence that the externality magnitudes are substantial, but none of this work delivers a calibrated Ē the paper would commit to as a parameter. Theorem 5.7'' is therefore a conditional aggregate-Kaldor-Hicks / potential-Pareto improvement claim, not an actual-Pareto ordering, and the empirical magnitude of E(·) relative to Ē is the open question the architectural case must eventually answer. The formal-model result stands without simulation backing; the empirical backing is correspondingly weaker.

5.8 Uniqueness and Comparative Statics

The EIMTS equilibrium is unique under standard gross-substitutes conditions and a bound on the social-welfare CES parameter λ_W. Proposition 5.5* (starred to distinguish from Theorem 5.5, the modified First Welfare Theorem): the EIMTS equilibrium is unique if gross substitutes hold and ∂λ_W/∂δ_L < κ*, where λ_W is the social-welfare CES parameter (eq. A.34; see Appendix A.1 for the notation glossary), δ_L is the central bank's discount factor in the loss function (eq. 1.1), and κ* is the elasticity-of-substitution-derived critical bound from the money-supply feedback Ψ(E). The condition states that the implied sensitivity of the welfare CES to the planner's discount factor must lie below the critical bound; this rules out a regime in which long-horizon welfare weighting feeds back through Ψ(E) to generate multiple equilibrium fixed points. The gross-substitutes assumption ensures excess demand slopes downward; the additional bound on ∂λ_W/∂δ_L ensures the money-supply feedback Ψ(E) does not generate multiple fixed points. For typical calibrations (the implied ∂λ_W/∂δ_L sensitivity in the empirically relevant range, κ ≈ 5–8 from the trade elasticity, κ* in the same magnitude order), uniqueness holds.

Comparative statics on the equality target: when G* tightens (the government targets lower inequality), the feasible-allocation set shrinks and feasible allocations become more equal, which requires redistribution. The price of labor rises (dp_l / dG* > 0); the price of capital services falls (dp_c / dG* < 0); the interest rate falls through the augmented rule's γ(G_t − G*) term. The trade adjustment runs symmetrically: tightening G* domestically raises the equality index E_ij vis-à-vis partner nations with looser targets, increasing the multiplier E_ij^ζ in the gravity equation and expanding bilateral trade volume. The system thus generates positive externalities in trade for nations that tighten their distribution targets, providing an institutional incentive for the kind of unilateral within-coalition equality improvement that the AGV mechanism rewards in expectation.

Comparative statics on λ_M (the money-supply equality sensitivity): higher λ_M produces stronger monetary response to equality changes; when actual equality exceeds target (E > E*), higher λ_M causes more monetary expansion, raising inflation but stimulating output and employment. The net effect on inequality is negative (E decreases toward E*), creating a stabilizing feedback loop. Comparative statics on γ (the Taylor-Rule equality coefficient): higher γ tilts policy toward inequality reduction; di* / dγ = (G − G*), so when G > G*, raising γ produces more aggressive rate cuts, stimulating wage growth through the labor-market channel. Inequality unambiguously falls (dGini / dγ < 0). The welfare effect is positive whenever the efficiency cost of the tighter feedback is small relative to the distributional gains — which it is, under standard calibrations.

5.9 The EIMTS Framework as Architecture

This chapter has presented the unified EIMTS model. The four instruments — equality-augmented Taylor Rule (§5.1), equality-indexed money supply (§5.2), equality-adjusted gravity model (§5.3), fair exchange-rate monitoring band (§5.4) — sit inside a constrained general-equilibrium problem whose formal specification appears in Appendix A. Equilibrium exists (Theorem 5.4'); the equilibrium is constrained Pareto optimal (Theorem 5.5); the second-welfare-theorem analogue holds as a non-operational existence result (Theorem 5.6'); the constrained equilibrium delivers an aggregate (Kaldor-Hicks) welfare improvement over the unconstrained competitive equilibrium when the inequality externality clears the lower bound Ē (Theorem 5.7''); the equilibrium is unique under standard gross-substitutes conditions (Proposition 5.5*). The architecture demonstrates that equality need not conflict with efficiency at the policy-relevant horizon and that the conventional separation of monetary policy from distributional concerns is theoretically unjustified — a point HANK has made within the heterogeneous-agent framework, and that the architectural-level EIMTS specification makes at the level of rules and institutions.

Implementation requires coordination among central banks, international financial institutions, and national governments. Chapter 7 develops the transition path and the institutional architecture required to make this formal model operational; Chapter 6 first presents the empirical anchoring through three case studies and the pre-specified econometric designs that the architecture's testable predictions admit.

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§ VIEmpirical Validation

The architecture developed in Chapter 5 is, on the page, a constrained general-equilibrium specification with four instruments and a welfare-superior equilibrium. On the ground, no part of it has been implemented at the scale the model contemplates. The empirical work of this chapter is therefore necessarily a work of partial validation: locating real-world institutional arrangements that implement one of the EIMTS instruments under conditions that expose what each does and what it cannot do without the rest of the architecture. Three cases serve the purpose. Bhutan's Gross National Happiness framework operationalizes a multidimensional equality target in the way the composite index of §5.2 contemplates. New Zealand's Wellbeing Budget translates a similar multidimensional measure into fiscal-allocation rules that mirror the f* operator of Theorem 4.7'. Mondragon Corporation's cooperative-federation structure implements the AGV-style intercooperation transfers of §4.4 within a single firm's institutional boundary. Each case is treated as a partial implementation under unfavorable conditions — none has the monetary, trade, and federation infrastructure EIMTS requires — and each is therefore informative both about what works and about why the full architecture would do more than the partial implementation does.

Chapter 6 does not report CGE simulation results. A February 2026 internal code audit identified five unresolved critical bugs in the simulation pipeline — a hardcoded stability Jacobian, an unenforced equality constraint in the equilibrium solver, a money-supply integration that computes but does not feed back, a sign error in the Gini dynamics, and a broken welfare-aggregation routine — and retaining bug-disclaimed outputs would preserve the appearance of empirical backing while admitting the backing was unsupported. The chapter is therefore organized as §6.1 strategy → §§6.2–6.4 case studies → §6.5 econometric designs (which carries the WID-vs-survey rank-correlation finding that is robust to the broken solver as a brief calibration-robustness observation) → §6.6 synthesis (which incorporates the audit acknowledgment as the closing context). The audit report and the prior replication archive remain bundled with the paper as an open research artifact pending bug-fix work; researchers building on the architecture should treat the prior simulation outputs as illustrative pre-fix outputs only.

6.1 Empirical Strategy

The empirical claim EIMTS makes is that an architecture with explicit distributional terms in monetary and trade rules produces measurably more equal outcomes than an architecture without them, at acceptable cost to aggregate output. The claim is multi-level — it operates at the level of national institutions, at the level of within-coalition trade, and at the level of global between-country distribution. No single empirical strategy can validate all three levels simultaneously. The case studies validate the institutional-level mechanism by showing that real institutions with one EIMTS-like instrument produce measurable distributional improvements within their scope of action. The econometric specifications in §6.5 provide the panel-data, instrumental-variables, and difference-in-differences specifications under which the EIMTS hypothesis would be testable as actual implementations begin to appear; that section also reports one calibration-robustness observation salvaged from the simulation pipeline (the WID-vs-survey rank-correlation finding) that survives the implementation bugs because it depends on the input data rather than the broken solver. The empirical strategy is therefore one of partial triangulation: no single piece of evidence is conclusive, but the convergence of partial-implementation case studies and pre-specified econometric tests is informative even in the absence of the CGE simulation outputs the bug-fix work has withheld.

6.2 Case Study I: Bhutan and Gross National Happiness

Bhutan adopted Gross National Happiness as a constitutional development objective in 2008 following more than three decades of explicit articulation as a national philosophy. The GNH framework operationalizes the multidimensional approach: nine domains (psychological wellbeing, health, education, time use, cultural diversity, good governance, community vitality, ecological diversity, living standards) are weighted using the Alkire-Foster (2011) methodology to produce a composite index S(p) = (1/k) Σ_d w_d · I_d(p) on each individual; aggregation across the population produces the national index. Pennock and Ura (2011) and Ura, Alkire, Zangmo, and Wangdi (2012) provide the technical methodology and the longitudinal data. The 2010, 2015, and 2022 GNH Surveys show measurable improvement on six of the nine domains, with the cross-survey "happy" share rising from 40.8% (2010) to 48.1% (2022); income inequality (Gini coefficient on disposable income) fell from 0.378 (2007) to 0.290 (2022), a reduction of 8.8 percentage points over 15 years.

The EIMTS-relevant feature of Bhutan is the institutional binding. GNH is not advisory: every government policy must pass a GNH Policy Screening Tool that scores it against the nine domains; policies scoring below the threshold are revised or rejected before submission to the National Assembly. The screening tool implements, at the policy-evaluation level, the same logic that EIMTS §5.5's social welfare function W(u_1, …, u_n) implements at the allocation-evaluation level. The isomorphism is direct: S(p) → f(θ_GNH), where θ_GNH parameterizes permissible allocation inequality. Bhutan demonstrates that a multidimensional equality measure can be institutionally binding at the policy-formation stage rather than only at the policy-evaluation stage, and that the binding can be sustained politically across multiple government turnovers (the 2008, 2013, 2018, and 2023 elections all retained the GNH framework despite partisan change).

What Bhutan does not demonstrate, and what the EIMTS architecture is designed to address, is the scaling problem. Bhutan's population is 770,000; its GDP is approximately $3 billion. The monetary and trade dimensions of EIMTS — which require coordination across central banks of larger economies, which require trade-cost adjustments operating through gravity-model multilateral resistance terms — do not appear in the Bhutan case because Bhutan's monetary architecture is a peg to the Indian rupee and its trade is overwhelmingly bilateral with India. The case validates the intra-national implementation of the multidimensional equality target as a binding constraint on policy; it cannot validate the international monetary and trade architecture EIMTS additionally requires. EIMTS formalism addresses the international layer that the Bhutan case structurally cannot reach. The case also has the obvious limitations of small-state generalizability: institutional commitments sustained in a 770,000-person constitutional monarchy may not survive transposition to a 330-million-person republic with multi-party political turnover, and the GNH framework's reliance on culturally specific psychological-wellbeing measures may not transpose to populations with different baseline expectations about state-citizen relations.

6.3 Case Study II: New Zealand's Wellbeing Budget

New Zealand adopted the Wellbeing Budget framework in 2019 under the first Ardern government, with the Treasury's Living Standards Framework (LSF) providing the technical infrastructure. The LSF identifies twelve domains of wellbeing and routes budget allocations through an explicit multidimensional optimization: B(w) = argmax W(x) subject to Σ x_i ≤ R, where W(x) is the wellbeing function over the twelve domains, x_i is the allocation to domain i, and R is total budget. The 2019, 2020, 2021, and 2022 budgets each had announced wellbeing priorities — child wellbeing, mental health, Māori and Pasifika wellbeing, environmental sustainability — with named appropriations attached. The framework survived the 2023 election change and continues under the Luxon government, though with reduced rhetorical emphasis.

The EIMTS-relevant feature of the Wellbeing Budget is the explicit instantiation of f* — the EIMTS allocation rule of Theorem 4.7' — as a fiscal-allocation mechanism. The Treasury's LSF specifies, in operational detail, how a multidimensional wellbeing target translates into actual budget line items. The mapping mirrors the EIMTS rule f*(θ) = {y_i | Gini(y, θ) ≤ Ḡ, Σ y_i = Y}: select allocations to maximize multidimensional welfare subject to the Gini-equivalent equality constraint and the aggregate-budget constraint. New Zealand's contribution is to demonstrate that this allocation rule is implementable at the level of a national fiscal cycle — that the Treasury, the cabinet, and the parliamentary appropriations process can be redesigned to operate against a multidimensional welfare function rather than against the implicit utilitarian-aggregate objective of standard public-finance budgeting.

The measurable outcomes are mixed and lagged. Child poverty (measured by the 50%-of-median post-housing-costs indicator) fell from 18.0% (2018) to 13.7% (2022), against a stated target of 10% by 2027; the rate has stabilized but not continued falling under the change of government. Mental health service access improved modestly; the housing crisis worsened across the same period despite explicit appropriation. The mixed pattern is instructive: the multidimensional allocation rule produces measurable progress on dimensions where the fiscal lever is available and binding, and no progress on dimensions (like housing) where the constraints are not primarily fiscal. EIMTS formalism addresses this limitation: the housing case shows that fiscal allocation alone is insufficient when the binding constraints are on the supply side (land-use restrictions) or on the monetary side (interest rates affecting construction finance); the four-instrument EIMTS architecture would route the monetary side through §5.1 and §5.2 in a way that single-instrument fiscal implementation cannot.

The case's limitations are political-economy in character. The Wellbeing Budget survived one election change but with substantially reduced commitment under the new government; the Treasury's LSF infrastructure remains intact but the political signal that it is the binding objective rather than a reporting framework has weakened. The EIMTS architecture's incentive-compatibility property (Theorem 4.7') is designed to prevent exactly this kind of unilateral policy reversal — once a nation enters the AGV-transfer system, defection costs the discounted value of future transfers — but no current real-world implementation provides that institutional binding. The case demonstrates the feasibility of the allocation rule and the political fragility of a non-incentive-compatible implementation; EIMTS adds the binding the Wellbeing Budget does not have.

6.4 Case Study III: Mondragon Corporation

Mondragon Corporation is a federation of approximately 80 cooperative enterprises based in the Basque region of Spain, employing roughly 70,500 worker-owners across industrial, retail, financial, and educational segments, with industrial-and-distribution sales of approximately €11.2 billion in 2024 (Mondragon Corporation, 2024; TU Lankide, 2024). The federation underwent a significant restructuring in 2022 when two large member cooperatives — ULMA Group (scaffolding and construction) and Orona (elevators) — voted to leave the federation, reducing the headline workforce by approximately 13 percent and revenue by approximately 15 percent; earlier reported figures of approximately 80,000 worker-owners reflect the pre-restructuring count. The federation was founded in 1956 by José María Arizmendiarrieta as a single small workshop and grew over six decades into Spain's seventh-largest corporate group. Whyte and Whyte (1991) and Cheney (1999) provide the institutional history; the contemporary financial reports (Mondragon Corporation, 2024) provide the operational data.

The EIMTS-relevant feature is the institutional implementation of the AGV-style transfer mechanism at the federation level. Three structural features matter. First, the pay-ratio constraint: each cooperative's bylaws set a maximum ratio w_max / w_min ≤ ρ where ρ ∈ {3, 4, 6} depending on the cooperative, producing an internal Gini coefficient G_M ≈ f(ρ) of approximately 0.22–0.28 depending on the skill distribution. The pay-ratio rule is the cooperative-level analogue of the EIMTS Gini constraint G(x) ≤ G* of §5.5. Second, the intercooperation funds: each cooperative contributes 10% of pre-tax surplus to a federation-wide solidarity fund (the Fondo de Educación y Promoción Cooperativa and related funds) which finances new cooperative formation, R&D, and bailout assistance to cooperatives in crisis. The intercooperation transfers implement, within a single legal-institutional boundary, the AGV mechanism's redistribution from successful to struggling coalition members. Third, the internal banking — Caja Laboral, founded 1959 — provides cooperative-rate credit r_coop = r_market − δ(E_coop − E*) where E_coop is cooperative employment and E* is a target, implementing the EIMTS equality-indexed money supply at the federation level.

The transfer rule the federation operates is a real-world instance of (4.AGV): Transfer_i = [r_market · K_i] − [r_coop(E_i, E*) · K_i], aggregated across all cooperatives; the difference between the market interest rate cooperatives would face externally and the equality-indexed rate they receive internally is the AGV-style transfer in expected-externality form. The Mondragon record is informative: across the 2008 global financial crisis, member cooperatives that would have failed in standard credit markets survived through Caja Laboral support, and the federation's overall employment fell less than the Spanish industrial-employment average over the same period. The 2013 collapse of Fagor Electrodomésticos — Mondragon's largest cooperative at the time — is the case's most-instructive limitation: when intercooperation transfers were insufficient to keep a 5,600-employee cooperative solvent against external shocks, the federation's solidarity mechanisms had to choose between sustaining Fagor through deepening debt and accepting its closure with redeployment of workers across other cooperatives. The federation chose redeployment, and 1,800 of Fagor's workers were absorbed into other Mondragon cooperatives within eighteen months.

What Mondragon demonstrates is that the AGV-style mechanism is institutionally implementable at the scale of a ~70,500-person federation (and previously, before the 2022 ULMA/Orona departures, an 80,000-person federation), that the implementation provides measurable crisis resilience, and that the inequality-reduction property holds (the federation's Gini is in the 0.22–0.28 range, substantially below the Spanish national Gini of approximately 0.33). What Mondragon cannot demonstrate is that the mechanism scales to the inter-national level the EIMTS architecture targets: Mondragon operates within a single legal-institutional boundary, with shared culture and shared exit costs, while EIMTS contemplates coordination across sovereign nations with heterogeneous institutions and substantial exit options. EIMTS formalism addresses the cross-jurisdictional layer that Mondragon, by virtue of operating within a single legal jurisdiction, structurally cannot reach. The federation's bylaws can be enforced through internal cooperative governance; EIMTS's analogous rules must be enforced through the AGV mechanism's incentive-compatibility property. The case validates the institutional feasibility of the transfer mechanism; the mechanism design of Chapter 4 provides what Mondragon's single-jurisdiction structure does not need to address.

6.5 Econometric Specifications

The testable predictions of the EIMTS architecture admit several econometric specifications under which the hypothesis becomes falsifiable as actual implementations begin to appear. The three specifications below are pre-registered designs that the paper proposes for future empirical work.

Panel data model with distributional outcomes. For nations i and years t, the relationship between monetary policy, trade integration, and the disposable-income Gini coefficient is

G_it = α_i + β_1 r_it + β_2 T_it + β_3 (r × T)_it + γ X_it + ε_it (6.9)

where r_it is the policy rate, T_it is trade openness, the interaction (r × T)_it captures monetary-trade complementarity, X_it is a vector of controls, α_i is country fixed effects, and ε_it is the error term. The EIMTS hypothesis predicts β_3 > 0 in absolute terms: the interaction of monetary and trade openness has a non-zero effect on inequality, with the sign depending on whether the architecture includes distributional terms. Under the current architecture (no distributional terms in either rule), the interaction effect is predicted positive (more openness, more inequality at higher integration intensities); under EIMTS the interaction effect is predicted to weaken or reverse as the equality-indexed instruments engage.

Instrumental-variables strategy. Endogeneity of monetary policy and trade with respect to inequality is the obvious econometric concern. The EIMTS architecture provides plausibly exogenous instruments: the announced monetary-policy reaction-function coefficients (which a central bank commits to) rather than the realized policy rates, and the committed trade-policy parameters (which a government signs into treaty) rather than the realized trade volumes. The instruments allow identification of the architectural-level effect separate from the realized-outcome endogeneity.

Difference-in-differences specification. For partial implementations — Bhutan's GNH adoption (2008), New Zealand's Wellbeing Budget (2019), Mondragon's intercooperation expansions — the difference-in-differences specification

Y_it = α + β(Treated_i × Post_t) + δ X_it + μ_i + λ_t + ε_it (6.10)

identifies the treatment effect of the implementation on the relevant outcome Y (Gini, child poverty, employment), with country fixed effects μ_i and time fixed effects λ_t. The synthetic-control method (as in Bhutan's case, where comparable Himalayan-region developing-economy controls are constructed weighted to match pre-treatment trends) provides robust identification under standard assumptions. Future EIMTS implementations would extend this specification to the cross-coalition level: nations entering EIMTS-style arrangements would be the treatment group, comparable non-participating nations the control, with the difference-in-differences estimator providing the causal effect of architectural-level adoption.

The data requirements are non-trivial: high-frequency panel data on disposable-income inequality at consistent definitions, harmonized trade and monetary policy variables across countries, and pre-registered analysis protocols to prevent post-hoc specification searches. The World Inequality Database (Alvaredo et al., 2018; Chancel et al., 2022) and the OECD's standardized inequality series (Solt, 2020) provide the inequality side; the IMF's monetary-policy database and the WTO's trade-policy database provide the institutional side.

Calibration-robustness observation. One finding from the prior CGE work survives the audit-identified solver bugs because it depends on input data rather than equilibrium computation: across WID pre-tax-national-income calibrations and household-survey-based calibrations of the equality-index input vector, the rank ordering of nations by Equality Index is preserved with a Spearman rank correlation of approximately 0.9. The robust interpretation of this observation is methodological rather than substantive: the relative diagnosis — which nations are most and least equal under the EIMTS measurement framework — is robust to the WID-vs-survey methodological choice, even though the absolute levels can differ substantially. This is informative for the econometric specifications above: panel-data and difference-in-differences identification of treatment effects depends on within-country variation rather than cross-country levels, and the rank-correlation robustness suggests that pre-registered tests of the EIMTS hypothesis can use either WID or survey-based Gini measures without their treatment-effect estimates being mechanically driven by the measurement choice. Earlier magnitude claims (e.g., "WID widens spread approximately 35%", "80% within tolerance") are not salvaged — those quantities depended on the solver outputs the February 2026 audit invalidated.

6.6 Synthesis

The empirical case for EIMTS is modest. The three case studies validate the institutional-level mechanism at single-instrument scope: Bhutan demonstrates the binding implementation of a multidimensional equality target; New Zealand demonstrates the operational instantiation of the f* allocation rule at fiscal scale; Mondragon demonstrates the AGV-style transfer mechanism at federation scale. The pre-specified econometric tests of §6.5 provide the framework for falsification as future implementations begin to appear, and the calibration-robustness observation confirms that the rank ordering of nations by equality index is robust to the WID-vs-survey methodological choice. The full architectural-level validation requires actual EIMTS implementation, which is the subject of Chapter 7. The case studies establish the institutional plausibility of the components; the architectural case rests on Chapters 1–5; the empirical claim is that the institutional plausibility and the architectural argument together are sufficient grounds to enter the transition phase. Whether they are sufficient to complete the transition is what Chapter 7 takes up.

CGE audit acknowledgment. The February 2026 internal code audit (EIMTS_CGE_AUDIT_REPORT.md) identified five unresolved critical bugs in the simulation pipeline: a hardcoded stability Jacobian, an unenforced equality constraint in the equilibrium solver, a money-supply integration that computes but does not feed back, a sign error in the Gini dynamics, and a broken welfare-aggregation routine. The simulation results have been withheld pending bug-fix work; the audit report and the prior replication archive (EIMTS_Replication_v3.zip) remain bundled with the paper as an open research artifact. A corrected implementation is in development; when the audit-identified issues are resolved, the corrected archive (EIMTS_Replication_v4.zip) will be released with an audit log mapping each fix to the corresponding audit finding, and the quantitative analysis the bugged pipeline cannot support will be folded back into a future revision. The paper rests on the architectural case, the case studies, and the pre-specified econometric designs — not on simulation magnitudes.

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§ VIIThe Transition Path

The architecture this paper has developed is not, even at the level of formal specification, the kind of object that can be implemented by single-actor decision. EIMTS requires coordination across central banks of the major economies; coordination across the trade-policy infrastructure of the WTO, OECD, and national trade ministries; coordination across the international-finance infrastructure of the IMF, BIS, and regional development banks; and political support sustained across electoral cycles in each of the participating nations. Any one of those requirements would be a substantial coordination problem on its own. Together they constitute a problem of the same order as Bretton Woods 1944 — and the empirical record of the post-1971 international monetary system is, to a first approximation, the record of partial attempts to reconstruct what Bretton Woods built. The transition path this chapter develops is therefore not a policy proposal in the conventional sense. It is a sketch of what would have to happen for EIMTS to come into existence, the named institutional precedents from which the relevant moves can be drawn, and the political-economy analysis of the resistance such a transition would encounter. The argument is honest about its own constraint: the architecture would be adopted not by accumulated rational persuasion but by crisis-window opening — analogous to how Bretton Woods was adopted in 1944, how the Tobin tax has been periodically revived (without yet being adopted) since 1972, and how the OECD minimum tax was adopted in 2021 after fifteen years of patient infrastructure building.

The transition path is organized in three phases over fifteen years. Phase I (years 0–3) establishes the diagnostic infrastructure required for the architecture to be implementable. Phase II (years 3–8) implements the architecture in partial form within voluntary coalitions of CME economies. Phase III (years 8–15) extends the architecture toward global coverage with the federated coalition structure that Theorem 4.7''s heterogeneous-coalition Shapley caveat motivates. Each phase has a stability profile derived from the Lyapunov function (full derivation in Appendix A.6), each phase's transition risks are analyzed at the level of the principal political-economy obstacles, and each phase's institutional precedents are named. The chapter closes with an explicit acknowledgment that the timing of the transition is dependent on crisis windows we cannot predict.

7.1 The Optimal-Control Formulation

Before describing the three phases, the formal structure of the transition problem deserves brief statement. Let z(t) = (z_m(t), z_t(t), z_i(t)) denote the state vector of the global economy, with z_m the monetary state (interest rates, money supplies, exchange rates), z_t the trade state (tariffs, gravity-equation parameters, bilateral equality indices), and z_i the inequality state (between-country and within-country disposable-income Ginis). The control variable u(t) = (u_γ, u_{λM}, u_ζ, u_δ) selects the EIMTS instrument intensities — Taylor-Rule equality coefficient γ, money-supply equality sensitivity λ_M, gravity equality elasticity ζ, exchange-rate monitoring-band bandwidth δ — and the social planner's transition problem is

min ∫_0^∞ e^(−ρt) L(z(t), u(t)) dt subject to dz/dt = F(z, u)

where L is the loss function and F describes the system dynamics. The control variable u(t) listed above uses u_λ for the money-supply equality sensitivity instrument intensity; this is the path of λ_M (see §A.1). The Hamilton-Jacobi-Bellman characterization

ρV(z) = min_u [L(z,u) + ∇_z V(z) · F(z,u)]

formally captures the planner's optimal control as a function of state and value-function gradient. The HJB equation is stated here but not solved in this paper. The optimal-control formalism marks the problem — locating the EIMTS transition inside the standard machinery of dynamic-programming public-economics rather than treating it as ad hoc — rather than supplying its numerical solution. The illustrative trajectories that might be derived under standard calibrations (γ(t) rising from 0 in Phase I through approximately 0.10 by mid-Phase II to a target band 0.20–0.25 by full implementation, with similar S-shaped paths for the trade-equality and exchange-rate-monitoring instruments) are intuitive features of the controlled-system dynamics rather than calibrated forecasts. The Gini-convergence rate λ_conv ≈ 0.20 per year obtained under the standard calibration is a calibration artifact derived from the closed-loop matrix's spectral properties under one parameter setting, not an estimated convergence speed; the actual convergence rate depends on the political-economy timing of Phase II initiation and the realized parameter values at adoption, both of which the formalism cannot foreclose. The HJB equation is the right object for a paper that would solve the planner's transition problem; this paper marks the object rather than solving it.

7.2 Phase I: Diagnostic Transparency (Years 0–3)

Phase I builds the measurement infrastructure required for the EIMTS rules to be implementable. The central institution is an International Equality Monitoring Authority (IEMA) modeled on the BIS's monitoring role for cross-border banking. The IEMA's mandate is to compute and publish four classes of indicators: harmonized Gini coefficients for each participating nation (using a shared definition that resolves the WID/survey discrepancy by reporting both); the residual Gini conditional on global state Z_t that serves as the type input to Theorem 4.7''s AGV mechanism; the bilateral equality indices E_ij that enter the gravity equation of §5.3; and the equality-augmented Taylor Rule's recommended rate path γ(G_t − G*) for each participating central bank, published as advisory but not yet binding. The reporting infrastructure is the load-bearing institutional commitment of Phase I; without harmonized cross-country measurement, the AGV mechanism's incentive-compatibility property cannot be verified and the gravity equation's equality term cannot be operationalized.

The institutional precedent is the IMF's Article IV consultation regime, which has established harmonized cross-country reporting on monetary and fiscal aggregates since 1944. The IEMA would extend this regime to inequality reporting, with the same combination of voluntary cooperation, technical-assistance backing, and reputational sanction for non-compliance. The political-economy challenge of Phase I is the conventional resistance to comparative inequality reporting — nations with high inequality have political incentives to obscure it, and statistical agencies have institutional incentives to use definitions that produce lower-Gini readings. The IEMA's mandate to publish multiple harmonized definitions (WID and survey) addresses this directly: nations that prefer one definition cannot suppress the other.

Phase I stability is straightforward. The system has not yet been augmented with the equality terms in any policy rule, so the dynamics are those of the existing global economy with one new advisory institution. The stability proof is trivial: the architecture has not changed, only the measurement layer has. The political risk is correspondingly modest: governments that disagree with IEMA reporting can publicly contest individual estimates without disrupting the broader institutional commitment, exactly as governments routinely contest IMF Article IV characterizations without exiting the Fund.

7.3 Phase II: Partial Implementation (Years 3–8)

Phase II implements the EIMTS instruments at partial intensity within voluntary coalitions of CME economies — the Wellbeing Economy Governments partnership (currently Scotland, New Zealand, Wales, Finland, Iceland, Canada), potentially expanded to include Sweden, Denmark, Norway, Costa Rica, and Bhutan. The phase consists of three parallel implementation tracks.

Monetary track: shadow equality rate. Participating central banks adopt a blended Taylor Rule i(t) = (1 − w(t)) · i_standard(t) + w(t) · i_EIMTS(t), with blending weight w(t) = min(t / T_blend, 1) and T_blend ≈ 5 years. The augmented rate i_EIMTS includes the γ(G_t − G*) term at intensity γ(t) = γ* · (1 − e^(−λ_conv t)), rising from 0 at Phase II start to approximately γ* = 0.10 by mid-Phase II (λ_conv per the §A.1 glossary disambiguation, here interpreted as the buildup-rate analogue of the closed-loop convergence rate). The blended rate becomes the actual policy rate; the standard-rule rate continues to be reported as the counterfactual baseline. This implementation pattern — gradual introduction of a new policy parameter with the old parameter retained for diagnostic comparison — is the standard institutional mechanism for monetary-policy regime transition (the inflation-targeting regimes adopted across the 1990s in New Zealand, Canada, the UK, Sweden, Australia followed this pattern).

Trade track: equality-adjusted trade preferences. Participating nations modify tariffs to τ_ij(t) = τ_base · (1 − δ · E_ij(t)), with the equality factor δ rising from 0 to approximately 0.20 over Phase II. The modification is most easily implemented inside existing preferential trade agreements (the EU's Generalized System of Preferences provides the institutional analogue) by adding the equality factor to the existing preference schedules. Nations with higher bilateral equality indices receive lower tariffs; the modification is unilateral on the part of the participating coalition, requiring no WTO-level negotiation in Phase II.

Exchange-rate track: fair-value monitoring band introduction. Participating nations implement the §5.4 monitoring band in its default-bandwidth form (δ_ij ≈ 0.30 around fair value, non-binding under the trilemma reframing). The band is bilateral within the coalition initially; nations outside the coalition continue to operate under the standard floating-rate regime against participating-nation currencies. The institutional precedent is the IMF's Article IV surveillance framework and the EU's six-pack macroeconomic-imbalances reporting, both of which publish benchmark-deviation information without triggering automatic intervention. The European Exchange Rate Mechanism is the binding-corridor version that the Mundell-Fleming trilemma rules out under open capital accounts; the monitoring-band version is the trilemma-compatible counterpart the architecture commits to. Phase II's exchange-rate track is therefore reporting-and-surveillance rather than binding-corridor; a binding upgrade is a Phase III possibility only if coordinated CFM implementation per §7.6 enables it.

Phase II stability is the load-bearing technical claim of the transition. The Lyapunov function V_2(z) = (1/2) z'Pz with positive-definite P satisfies dV_2/dt ≤ −λ_min(Q) · ||z||² where Q is the symmetric part of the augmented-system matrix. The technical details are in Appendix A.6; the conclusion is that Phase II is stable for any γ < γ̄ ≈ 1.70 (well above the proposed γ* = 0.10 implementation intensity). Stability holds independently of which other coalitions join, because the partial implementation is internally consistent within the coalition's monetary-trade boundary.

Phase II's political-economy challenge is the coalition coordination problem at the moment of binding commitment. Each participating central bank must adopt the blended Taylor Rule simultaneously, because unilateral adoption by one central bank without coalition support produces exchange-rate misalignments that disadvantage the adopter. The coordination mechanism is a binding multilateral treaty similar in form to the 1985 Plaza Accord (the G5 coordinated dollar devaluation), with the IEMA serving as the technical-monitoring authority for compliance. The Plaza Accord is the most-recent precedent for coordinated monetary-policy action across major economies, demonstrating that the institutional capacity for the coordination exists; the Plaza Accord's narrow scope (one-time devaluation) compared with EIMTS's broader scope (ongoing rule augmentation) marks the size of the institutional ambition.

7.4 Phase III: Full EIMTS (Years 8–15)

Phase III extends the architecture toward global coverage, with the equality intensity γ rising to its calibrated welfare-optimal level (approximately 0.20–0.25 in advanced economies), the gravity equality elasticity ζ stabilizing at approximately 0.20, and the exchange-rate monitoring band reaching its full reporting scope (the band itself remaining non-binding under the trilemma compliance flagged in §5.4, unless coordinated CFM implementation per §7.6 enables a binding upgrade). The phase's distinguishing feature is the federated coalition structure that Theorem 4.7''s Shapley caveat motivates: rather than a single flat coalition of all participating nations, EIMTS in Phase III operates as a federation of size-homogeneous coalitions (the advanced-economy coalition, the large-developing-economy coalition, the small-open-economy coalition, with regional sub-coalitions in each), linked by a top-level coordinating layer based at a reformed IMF. The federation structure addresses both the Shapley-value heterogeneity concern (within each coalition, GDP ratios are bounded; across coalitions, the federation layer manages the asymmetry) and the political-feasibility concern (each coalition can adopt at its own pace, and full global coverage is reached by aggregation rather than by simultaneous global adoption).

The institutional precedent for the federation structure is the European Stability Mechanism (ESM, 2012) layered on top of the European Central Bank, with the IMF playing the federation-level role. The ESM provides a partial template for the inter-coalition transfer mechanism: nation-level central banks operate within their coalition's monetary regime, with the federation-level institution providing transfer-mechanism implementation and last-resort financial support. The transition from coalition-level partial implementation to federation-level full implementation is the institutional analogue of the transition from the ERM to the euro, but with the explicit constraint that EIMTS does not impose a single currency — coalitions retain currency sovereignty, and the fair-value monitoring band of §5.4 operates as a transparency reporting layer rather than as a binding intervention regime in the absence of coordinated CFMs.

Phase III's political-economy challenges are substantially harder than Phase II's. Three obstacles deserve named attention.

Central-bank-independence orthodoxy. The intellectual case for central-bank independence (Rogoff, 1985) rests on the time-inconsistency argument: democratically-elected governments will accept inflation today to gain employment today, deferring the inflation cost to a successor government, so central-bank monetary policy is delegated to an institution shielded from short-term political pressure. The EIMTS architecture adds a distributional response to the central bank's mandate, which the orthodoxy reads as politicizing the institution. The response is that the EIMTS distributional response is rule-based rather than discretionary: the augmented Taylor Rule (eq. 5.1) constrains the central bank to a published reaction function on a multidimensional set of aggregates, exactly as the standard Taylor Rule constrains it to a reaction function on inflation and output. Rule-based distributional response is institutionally distinct from discretionary fiscal redistribution; the central bank's independence from day-to-day political pressure is preserved. The orthodoxy is empirically defensible at its strongest point (avoiding the inflation-bias of discretionary regimes) and weakest at its inference (that any non-aggregate target politicizes the institution); the EIMTS architecture's rule-based structure addresses the strong point and avoids the weak inference.

IMF/BIS conservatism. The international-finance institutions have institutional cultures that prize macroeconomic stability and resist innovations that might create new sources of instability. The conservatism is functional — the institutions' role is to provide stability backstop during crises, and they cannot afford to be sources of instability themselves. The EIMTS architecture's transition path must therefore satisfy a substantially higher institutional-credibility bar than its theoretical case alone would justify. The credibility argument runs through the case-study evidence of §§6.2–6.4: each EIMTS instrument has been implemented in partial form at smaller scale without producing instability, and the federation structure of Phase III lets the IMF/BIS-level coordination layer build on the demonstrated stability of the smaller-scale implementations rather than betting the institutional reputation on untested full-scale adoption. The institutional-credibility argument is, in effect, that EIMTS is the outcome of fifteen years of incremental institutional construction rather than a single discrete adoption event.

Financial-sector lobbying. The financial sector — banks, asset managers, hedge funds, and the institutional investors collectively holding the world's portfolio assets — has well-documented political influence in OECD economies (Acemoglu & Robinson, 2019). The EIMTS architecture's monetary-side instruments (the augmented Taylor Rule, the equality-indexed money supply, the seigniorage-redistribution mechanism implied by the AGV transfers) directly reduce the asset-price channel through which the sector benefits from current monetary architecture. The sector therefore has straightforward incentives to resist EIMTS adoption, and the historical record of comparable reforms (Glass-Steagall repeal 1999, Dodd-Frank 2010, the post-2008 wave of macroprudential regulation) suggests that financial-sector lobbying is effective at delaying and weakening such reforms without preventing them outright. The political-economy strategy is to time EIMTS adoption to crisis windows when the sector's political influence is temporarily weakened — the post-2008 reform window, narrow as it was, demonstrates the pattern.

Phase III stability follows from the Lyapunov analysis with the full instrument set: V_3(z) = (1/2)[(π−π*)² + α_1(y−y*)² + α_2(G−G*)² + α_3(e−e*)²] decreases along the system trajectory under standard conditions (dV_3/dt = −z'Qz with Q positive-definite for the empirically relevant parameter range). The system's local convergence to z* with ||z(t) − z*|| = C · e^(−λ_conv t), λ_conv ≈ 0.20 per year (per the §A.1 glossary disambiguation), means full convergence within approximately 7 years of Phase III initiation, completing the architecture's installation around year 15 of the transition.

7.5 Precedents and Coordination Capacity

Three institutional precedents are load-bearing for the transition argument. Each demonstrates that the order of magnitude of coordination EIMTS requires has been achieved before, under specific conditions worth naming.

Bretton Woods, 1944. The most-cited precedent for international monetary architecture creation is the simultaneous adoption of a managed-exchange-rate system, the establishment of the IMF and World Bank, and the negotiation of GATT principles, achieved at a single conference of forty-four nations over three weeks in July 1944. The institutional context that made Bretton Woods possible — the immediate post-WWII rebuilding requirement, the unambiguous US economic and military dominance enabling coordination from a single hegemonic position, the prior intellectual preparation in the Keynes and White plans of 1942–1943 — is unrepeatable. But the scope of the coordination achieved (monetary, trade, and development-finance architecture in a single multilateral commitment) is the order-of-magnitude precedent for EIMTS. The lesson is that comparable architectural commitments are possible at moments of acute crisis, with substantial intellectual preparation in advance.

The Tobin tax debate, 1972–present. Tobin (1972) proposed a small tax on foreign-exchange transactions to discourage destabilizing speculation. The proposal has been periodically revived (notably during the late-1990s Asian financial crisis and after the 2008 global financial crisis) without being adopted. The lesson is twofold. First, theoretical proposals for transactional reform of the international financial system can survive for decades without adoption — the Tobin tax has been on the international policy agenda for over fifty years. Second, the absence of adoption reflects coordination-problem dynamics rather than theoretical refutation: the case for the tax has been incrementally strengthened by subsequent research (Eichengreen and various co-authors across the 1990s; the post-2008 macroprudential literature) without producing the political coordination required for implementation. EIMTS is at the analogous point: theoretically specified, intellectually defensible, awaiting the political-coordination window.

OECD minimum tax, 2021. The most-recent successful coordination on inequality-relevant ground is the 2021 OECD agreement to implement a 15% global minimum corporate tax rate, signed by 137 jurisdictions and progressively implemented across 2023–2025. The agreement followed approximately fifteen years of OECD Base Erosion and Profit Shifting (BEPS) infrastructure building, demonstrating that a sustained multilateral commitment to inequality-relevant coordination is achievable in the contemporary institutional environment. The lesson for EIMTS is that the IEMA-as-Phase-I institution-building can serve the same infrastructure-development role for the broader EIMTS coordination that BEPS served for the minimum tax: patient, technical, multilateral preparation that produces the institutional readiness for adoption when the political window opens.

7.6 Transition Risks and the Capital-Controls Structural Complement

The risk inventory is short and explicit. The numerical values reported below are illustrative rather than estimated — they reflect the structure of the underlying risk-matrix exercise, not a calibrated probabilistic forecast. Capital flight from coalition nations during Phase II transition is a moderate-probability, moderate-impact risk (illustratively, on the order of 30% / moderate), mitigated by transparent communication, the structural capital-controls complement discussed below, and federation-level liquidity support. Metric gaming (nations misreporting their inequality status to capture AGV transfers) is a lower-probability, lower-impact risk, mitigated by multi-source verification through the IEMA, independent audits, and the AGV mechanism's incentive-compatibility property itself (which makes truthful reporting individually optimal). Coordination failure in Phase II — one participating central bank exits during a domestic political reversal — is a lower-probability, higher-impact risk, mitigated by exit clauses with cost provisions, supermajority voting requirements for coalition decisions, and the network effects that make participation more valuable as the coalition grows. Political reversal at the national level is an ongoing per-year risk with moderate-to-high impact, mitigated by embedding EIMTS commitments in binding international treaties, building domestic political coalitions through the case-study evidence of §§6.2–6.4, and structuring the AGV transfers to favor populations with broad political distribution. Measurement dispute over harmonized inequality estimates is a moderate-probability low-impact risk, addressed by the dual-calibration reporting infrastructure and an arbitration tribunal within the IEMA structure. The illustrative expected impact across the risk profile (E[Impact] ≈ 3.0 on a 10-point scale, computed as Σ_i P_i · Impact_i under the illustrative values) suggests the transition is risky but not catastrophic in expectation; the value is illustrative because none of the constituent probabilities is empirically estimated, and the scale itself is internal to the risk-matrix exercise.

Capital controls as structural complement. Forced by the §5.4 trilemma honesty, capital-controls coordination is a structural complement to the architecture rather than a transitional risk-management measure. The §5.4 monitoring band is non-binding because a binding version with open capital accounts would violate the Mundell-Fleming trilemma; the path to a binding fair-value corridor — if the political economy ever supports it — runs through coordinated capital-controls implementation across coalition members. The IMF's 2012 institutional view on capital-flow management (CFM) measures, and its 2022 update treating CFMs as a legitimate part of the macroprudential toolkit, provides the institutional precedent. Phase II coalition coordination would include CFM-coordination as a baseline commitment, not as an emergency response; the federation-level coordination layer of Phase III (§7.4) would extend the CFM-coordination scope to inter-coalition transfers.

The structural-complement framing has two consequences. First, the §5.4 corridor can be narrowed in Phase III implementation if and only if coalitions accept coordinated CFMs — the architectural choice trades exchange-rate predictability against capital-flow openness, and the architecture makes the trade-off explicit rather than presupposing one side of it. Second, the political-economy obstacle to capital controls is admitted directly: the post-Bretton-Woods consensus against capital controls is intellectually weaker than it was twenty years ago (the IMF 2012 institutional view marks the inflection), but the financial-sector lobbying analyzed in §7.4 has strong incentives against CFM coordination, and the political coalition for the structural-complement framing is correspondingly harder to build than the political coalition for the rate-rule-only architecture. Cross-reference back to §5.4: the non-binding monitoring band is the architecture's commitment in the absence of CFM coordination; the binding corridor is the architecture's reachable extension if CFM coordination is built in Phase II–III.

7.7 The Transition Theorem

We close the chapter with the formal statement that the transition path is convergent under the stated conditions. The full proof is in Appendix A.6.

Transition Theorem. Under the three-phase transition path with Phase II Lyapunov stability (proven in Appendix A.6) and Phase III Lyapunov stability (similarly proven), the system state z(t) converges to the EIMTS steady state z* with an exponential rate determined by the closed-loop spectral properties of the controlled system. Under the illustrative calibration of Appendix A.6 the implied convergence rate is approximately λ_conv ≈ 0.20 per year — labeled here explicitly as a calibration artifact rather than an estimated convergence speed, since it depends on parameter values (the implied semi-elasticities in the closed-loop matrix) the paper does not estimate from data. The convergence is robust to perturbations of magnitude bounded by the stability margin λ_min(Q) > 0, where Q is the symmetric part of the closed-loop system matrix.

The substantive content of the theorem is that EIMTS, once started, will converge to its calibrated steady state under the stated stability conditions; the speed of convergence under any specific parameter realization is a matter of estimation rather than of architecture. Under the illustrative calibration above, full convergence would obtain within approximately 7–8 years of full Phase III initiation, completing the architecture's installation around year 15 of the transition; this should be read as an order-of-magnitude indicator of the policy-relevant timescale rather than as a forecast. The transition's timing depends on when Phase II is initiated — which depends on the political-economy conditions analyzed above — but once initiated, the trajectory is convergent under the stated stability conditions. Whether the timing condition is satisfied is what the next decade will determine. The architecture is ready; the institutional preparation is the patient work of Phase I; the implementation is the political work of Phases II and III. The conclusion of this paper reflects on whether the work is likely to be done.

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§ —Conclusion

This paper has not delivered a complete theory of distributional macroeconomics, nor a finished implementation of the architecture it specifies, nor any quantitative simulation result, since the simulation pipeline has been withheld pending bug-fix work. The honest accounting of what the paper does not do is the place this conclusion has to begin, before any claim about what it does do can be evaluated against the right baseline.

Five limitations bound the contribution. First, the computable general equilibrium implementation has been withdrawn rather than retained with disclaimer. The February 2026 internal audit identified five unresolved critical bugs in the simulation pipeline, and bug-disclaimed simulation outputs would have retained the appearance of empirical backing while admitting that backing was unsupported. The architecture-level claims rest on the formal results in Chapters 4 and 5, the partial-implementation case-study evidence of §§6.2–6.4, and the pre-specified econometric designs of §6.5 — not on simulation magnitudes. That is a substantial reduction in the empirical confidence the paper claims, and one a fair reader is entitled to discount the architectural claim against. The audit report and the prior replication archive remain bundled as an open research artifact; a corrected implementation is in development.

Second, the formal model relies on stylized rational-actor assumptions — log-normal income distributions in the Bridging Proposition, quasi-linear preferences in the AGV mechanism, the common-prior assumption underlying the BIC guarantee, and the maintained empirical hypotheses of r > g and incomplete fiscal offset in Theorems 1.1' and 3.1' — that the contemporary heterogeneous-agent macroeconomic literature (Kaplan, Moll, & Violante, 2018; Auclert, 2019; Bayer, Born, & Luetticke, 2024) has shown to be substantial simplifications. The HANK framework's finding that monetary policy's distributional effects operate through portfolio-revaluation channels concentrated at the top of the wealth distribution and through liquidity-constrained-household channels concentrated at the bottom is more granular than the architectural-level treatment of this paper supports. EIMTS as specified is a rule-based response to aggregate distributional measures; the HANK literature would suggest that more granular instrument targeting may be welfare-improving. The architectural simplification is defensible at the rule-design level — central banks operate on rules whose stability requires aggregate-level objects — but it is a simplification, and the aggregate welfare improvement of EIMTS over the current architecture, captured conditionally in Theorem 5.7'', may understate the welfare improvement a HANK-informed instrument design could achieve.

Third, the architecture as specified targets disposable income Gini rather than wealth Gini, and the rate-cut channel that improves the disposable-income Gini through labor-market stimulation simultaneously raises asset prices and so moves the wealth Gini in the opposite direction. The §5.1 wealth-Gini side-effect subsection proposes macroprudential complementarities (LTV/DSTI tightening, differential capital surcharges) as the off-rate instrument set that handles what the rate rule alone cannot, and §7.2's IEMA is extended to harmonized wealth-distribution reporting from Phase I to support this. But the joint architecture that specifies the income-and-wealth coordination at the rule level — the analogue of §5.1 for the macroprudential side — is open work that this paper does not deliver.

Fourth, the Atkinson constraint substitution that Theorem 5.4' commits to makes the Gini target G* a reporting metric rather than a formal binding constraint. The Bridging Proposition recovers the Gini-policy-target language for log-normal distributions, and the Lorenz-domination containment recovers it conservatively for general distributions, but the policy-side conversation about "Gini target 0.30" is technically about an Atkinson constraint A^ε(x) ≤ A* that corresponds to a Gini ceiling G† under the bridge. The policy operationalization of the architecture therefore needs to handle the distinction with care: implementation documents that present G* as the binding object will be vulnerable to challenge when actual distributions are not log-normal (which most actual distributions are not, in the tail Pareto-shape that Chapter 3 documented). The Atkinson constraint is the right formal object; the policy presentation needs to be honest about what is reported and what is binding.

Fifth, the architecture's flagship national instrument addresses within-country inequality, but the diagnosis that bites most pressingly is the between-country distribution (Ḡ ≈ 0.75–0.85 globally, versus the OECD-average Ḡ ≈ 0.33 within rich nations under the §3.7 calibration). The augmented national Taylor Rule is structurally a poor match for the between-country target, which is governed by growth-and-convergence dynamics across nations rather than by any single central bank's reaction function. §3.7's diagnostic-gap acknowledgment makes this explicit; the multilateral coordination architecture of §7 is where the between-country diagnosis would actually be addressed; the national rule's role is to ensure individual nations are not actively worsening the global picture while the multilateral architecture is constructed. This is a target-instrument mismatch the architecture surfaces explicitly.

Sixth — really, a continuation of the fifth — the transition path of Chapter 7 is contingent on political-economy conditions that are not in the architecture's control. The architecture is ready; the institutional preparation of Phase I is achievable through existing IMF/OECD infrastructure; the partial implementation of Phase II is within the institutional capacity of the Wellbeing Economy Governments coalition. But the global Phase III implementation depends on a crisis window analogous to those that opened for Bretton Woods (1944), the post-1971 monetary reorganization, the post-2008 macroprudential reforms, and the OECD minimum tax (2021). Such windows cannot be reliably predicted. The paper's argument is that the architectural and institutional preparation should be done in advance of the window, so that when it opens, the architecture is available to take up. Whether the preparation will be done is a question this paper poses; whether the window will open is a question this paper cannot answer.

Against those limitations, the paper's contribution is sharply bounded. Theorem 1.1' establishes that any monetary policy rule of the Taylor form embeds distributional asymmetry as an unconditional architectural feature, independent of parameterization. The empirical-consequential reading of that asymmetry is conditional on r > g and incomplete fiscal offset, both of which are testable rather than assumed. Theorem 2.4' establishes the trade-architecture analogue, weakened from "by logical necessity" to a joint-conditional inference from the five mechanisms under four enabling conditions. Theorem 3.1' establishes bounded logistic convergence of the Gini to an asymptote Ḡ < 1 whose magnitude depends on the strength of counterforces — and shows that the case for EIMTS is in reducing Ḡ below politically tolerable levels, not in preventing catastrophe. Theorem 4.7' establishes the AGV-mechanism-based incentive-compatible architecture for cross-national coordination, committing to BIC + IIR + ex-ante BB + ex-ante efficiency as the load-bearing guarantee; a DSIC strengthening is not claimed, because the five-dimensional EIMTS type space does not admit the Myerson one-dimensional payment identity without a defensible scalar projection. Theorem 5.4' establishes equilibrium existence under the Atkinson constraint, with the Bridging Proposition (standard A^ε = 1 − exp(−εσ²/2) form for log-normal distributions) recovering Gini-denominated reporting. Theorem 5.6' establishes the second-welfare-theorem analogue as a non-operational existence result — the required initial-endowment redistribution exceeds the seigniorage capacity of the architecture's monetary instruments. Theorem 5.7'' establishes that the constrained equilibrium delivers a conditional aggregate (Kaldor-Hicks) welfare improvement over the unconstrained baseline when the inequality externality clears a contested lower bound Ē; absent operational compensating transfers (which Theorem 5.6' has already told us are not operational), the improvement is potential-Pareto / aggregate-welfare rather than actual-Pareto at the household level.

The architectural-level contribution is to demonstrate that the absence of any distributional term in the world's two most consequential rule-following institutions is a design feature rather than a technical accident, and that an explicit equality instrument can be added to each without violating known impossibility results. The instrument additions are not trivial — they require coordination across central banks, treaty-level binding in international finance, and federation-level institutional architecture — but they are feasible at the level of formal specification. The paper's argument is that the architectural feasibility is itself an important result, because the standard framing of the inequality problem within the Tinbergen-separation orthodoxy treats distributional response as the province of fiscal policy alone, and the fiscal-policy response has demonstrably been insufficient to reach the inequality levels OECD nations articulate as their political objective.

The closing reflection ties to the broader theme that runs across this archive, articulated most clearly in Paper 003's "Living Room and Long Grip" and in the operating-system framing of Paper 004: that distribution is not a residual that markets clear toward but a primitive that institutions choose. The Beyond the Binary paper (Paper 001) makes the same point at the motivational level: institutions embed assumptions about human motivation, and those assumptions can be changed by institutional design rather than by exhortation. EIMTS makes the analogous architectural-level point about monetary and trade institutions: the assumption that distribution is exogenous to monetary and trade rules is itself a design choice, and the choice can be made differently. The choice the current architecture makes — that the Taylor Rule does not respond to distribution, that the gravity model does not respond to bilateral equality asymmetry, that the international monetary system does not monitor exchange-rate deviations against equality-adjusted fair-value benchmarks — is the choice that produces the empirical outcomes Theorems 1.1' and 2.4' describe. A different choice is implementable. Whether it will be implemented is what the next decade of architectural preparation, partial-coalition adoption, and crisis-window arrival will determine.

Whether the institutions of the next half-century will be designed around fair exchange, or merely around market-clearing exchange, is the question this paper poses; it is not the question this paper answers.

What this paper has shown

A sympathetic reader who has followed the conditionals — r > g maintained, Ē unestimated, ex-post BB fragile, common prior assumed, gravity reduced-form, wealth-Gini deferred, between-country target relocated to §7, i / M reduced from two independent levers to one two-aspect monetary lever, quasi-linearity bounded where transfers are not yet consequential — may reasonably wonder what positive content remains. The catalogue of boundaries should not displace the catalogue of contributions inside them. Five positive claims survive the limitations and are stated affirmatively here.

(a) Architectural feasibility. An equality term can be introduced into the world's two most consequential rule-following institutions — the central-bank policy rule and the gravity model of trade — without violating any of the impossibility results that govern those institutions. Theorem 4.7' establishes the architecture under BIC + IIR + ex-ante BB + ex-ante efficiency; Theorem 5.4' establishes equilibrium existence under the Atkinson constraint; Theorem 5.6' establishes the supporting allocation's existence at the formal level even where its operational redistribution is not financeable; Theorem 5.7'' establishes the conditional aggregate-welfare improvement over the unconstrained baseline.

(b) Impossibility mapping. The paper has shown precisely what Green-Laffont, Arrow / Sen, and Mundell-Fleming each permit and forbid in the architecture's design. The architecture's commitments are consistent with these impossibility results: BIC-not-DSIC respects Green-Laffont; conditional-Kaldor-Hicks-not-strict-Pareto and capability-cardinality-not-ordinal-preference respect Arrow-via-Sen; non-binding monitoring band respects Mundell-Fleming.

(c) Disposable-income-Gini commitment with macroprudential complementarity. The architecture is honest about which inequality it targets (disposable-income Gini at WID/LIS standards) and where the complementary instrument set sits (a macroprudential committee on the wealth-Gini side, outside the monetary-trade core, staged jointly with Phase II in §7). The off-rate instrument set — LTV/DSTI caps, differential capital surcharges, transaction-tax differentiation — is named in the §5.1 Tinbergen table and operationalized in the §7.2 IEMA's extended wealth-distribution reporting mandate.

(d) Between-country diagnosis relocated to multilateral architecture. The §7 Bretton Woods 2.0 / equality clearinghouse / OECD-equality-tax sketches are where the between-country inequality — the paper's most pressing diagnosis — would be addressed. The national rule is correctly understood as a within-country containment instrument rather than as a between-country-redistribution device; the multilateral architecture is where the between-country contribution lives.

(e) AGV-BIC mechanism core. The d'Aspremont-Gérard-Varet (1979) expected-externality construction, with explicit common-prior and ex-post-BB-fragility caveats and with the quasi-linearity-domain tension surfaced, is the formal core of the implementation mechanism. The mechanism does what BIC + ex-ante BB can do — align reporting incentives with social welfare under a coordinating institution that maintains the conditioning-set Z_t in shared methodology — and does not pretend to deliver the DSIC + ex-post BB combination Green-Laffont rules out.

The conditionals catalog the boundaries of what this paper has shown; the five points above catalog what it has shown within those boundaries. The argument is that distribution is not a residual markets clear but a primitive institutions choose, and that the institutional choices through which distribution becomes a primitive can be specified to a level of formalism that respects the impossibility results constraining such designs.

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The replication archive (EIMTS_Replication_v3.zip) corresponds to the pre-audit CGE code; see Appendix B.6 for details. Comments welcome at contactme​@​marshallcahill.com.

Appendix A: Formal Model and Proofs

This appendix consolidates the formal specification and the proof bodies that the main text deferred. The structure is: A.1 notation glossary; A.2 the EIMTS economy formal specification (Definition A.1, restating §5.5's content); A.3 social welfare functions; A.4 proof bodies for Theorems 1.1', 3.1', 4.7', 5.4'; A.5 additional proofs (Theorem 2.4' trade-structural; the modified welfare theorems 5.5/5.6'/5.7''); A.6 stability analysis (Theorem 5.1 derivation and the Lyapunov analysis for Chapter 7); A.7 the Bridging Proposition.

A.1 Notation Glossary

Several symbols carry potential overload; the glossary disambiguates the four λ-uses (loss-function output weight; money-supply equality sensitivity; social-welfare CES parameter; convergence rate / eigenvalue) and the two μ-uses (loss-function equality weight; entrepreneurial-disruption rate) into explicit subscripted forms.

• γ (gamma) — Taylor Rule equality response coefficient (eq. 5.1) only.

• ζ (zeta) — gravity model equality elasticity (eq. 5.15).

• β (beta) — Taylor Rule output coefficient (eqs. 1.2, 5.1) only.

• δ_L (delta-L) — central bank discount factor in the loss function (eq. 1.1).

• κ (kappa) — Piketty capital-to-income ratio (§3.4), following Piketty's own notation.

• λ_y (lambda-y) — loss-function output-stabilization weight in eq. (1.1).

• λ_M (lambda-M) — money-supply equality sensitivity in §5.2 and B.4 (eq. 5.9, Ψ(E_t) = exp(λ_M(E_t − E*))).

• λ_W (lambda-W) — social-welfare CES parameter in Proposition 5.5* and eq. (A.34).

• λ_conv (lambda-conv) — convergence rate / eigenvalue in the §A.6 Lyapunov analysis and the §7 Transition Theorem.

• μ_G (mu-G) — equality weight in the §5.1 loss-function rate rule modification.

• μ_d (mu-d) — entrepreneurial-disruption rate in the §3.6 Gini ODE.

• AGV — d'Aspremont-Gérard-Varet (1979) expected-externality mechanism (Theorem 4.7'). VCG appears only as the Green-Laffont impossibility foil.

• A^ε — Atkinson index with inequality-aversion parameter ε (eq. 3.6).

• A* — Atkinson constraint level used as the formal binding constraint in Definition A.1 and Theorem 5.4'.

• G* — Gini target as a reporting metric, recoverable from A* via the Bridging Proposition (§A.7).

• G_w — wealth Gini, distinct from the disposable-income Gini G; the macroprudential row of the §5.1 Tinbergen table addresses G_w.

• Ḡ — asymptotic Gini steady-state value in Theorem 3.1'.

• φ — fiscal offset ratio in Theorem 1.1' Conditional Corollary.

• Z_t — global state vector (g_t^W, p_t^W, i_t^W) used in the AGV residual-type definition (§4.4).

A.2 The EIMTS Economy: Definition A.1

Definition A.1 (EIMTS Economy). An EIMTS economy is a tuple E = (I, J, (ω_i), u_i, (Ω_j), Y_j, p*, i*, W, A*) where I = {1, …, n} is the set of households, J = {1, …, m} is the set of firms, ω_i is the initial endowment vector of household i in commodity and labor space, u_i: ℝ_+^L × [0,T] → ℝ is the utility function of household i, Ω_j ⊂ ℝ^L is the production set of firm j, Y_j is firm j's output, p* is the equilibrium price vector, i* is the equilibrium nominal interest rate, W is the social welfare function (Rawlsian, capabilities-based, or generalized CES), and A* is the target Atkinson index threshold.

The EIMTS equilibrium solves max W(u_1(x_1), …, u_n(x_n)) subject to:

(A.1.1) Market clearing: Σ x_i = Σ ω_i + Σ y_j

(A.1.2) Household budget: p* · x_i ≤ p* · ω_i + π_i + τ_i for all i, where π_i is firm-ownership profit share and τ_i is the seigniorage-financed transfer.

(A.1.3) Firm optimization: y_j ∈ argmax {p* · y : y ∈ Ω_j} for all j.

(A.1.4) Equality constraint: A^ε(x_1, …, x_n) ≤ A* — the Atkinson constraint as the binding object. The Bridging Proposition of §A.7 recovers Gini-denominated targets.

(A.1.5) Monetary policy rule: i_t = r* + α(π_t − π*) + β(y_t − y*) + γ(G_t − G*) — equality-augmented Taylor Rule (eq. 5.1).

(A.1.6) Money supply: M_t = M_t(base) · exp(λ_M(E_t − E*)) — equality-indexed money supply rule (eq. 5.9), with λ_M the money-supply equality sensitivity per the §A.1 glossary.

(A.1.7) International trade: T_ij = A · (Y_i · Y_j / Y_W) · (t_ij / (Π_i · P_j))^(1−σ) · E_ij^ζ — equality-adjusted gravity equation (eq. 5.15; the (1−σ) exponent matches the Anderson-van Wincoop form of eq. 2.27). The reduced-form-versus-structural caveat noted after eq. (5.15) applies here as well: the augmentation does not preserve the closed-form Anderson-van Wincoop fixed-point structure under endogenous multilateral resistance.

A.3 Social Welfare Functions

EIMTS admits multiple welfare specifications. The Rawlsian form W(u_1, …, u_n) = min_i u_i (eq. A.32) maximizes the welfare of the worst-off household. The capabilities-based form W(u_1, …, u_n) = min_i C_i where C_i = C(u_i, HDI_i, etc.) (eq. A.33) extends the maximin criterion to multidimensional capability sets (Sen, 1999). The generalized CES form W = (Σ w_i u_i^(2−ρ))^(2/(2−ρ)), where ρ ≡ λ_W is the CES substitution parameter related to inequality aversion by ρ = (1−ε)/ε (eq. A.34), interpolates between the utilitarian limit (ε→0, ρ→∞) and the Rawlsian limit (ε→1, ρ→0). Different welfare specifications generate different equilibrium allocations under the same Atkinson constraint; the comparative-statics analysis of §5.8 shows that the constrained equilibrium properties (Theorems 5.4', 5.5, 5.6', 5.7'') hold under all three specifications under standard convexity conditions.

A.4 Proof Bodies for Theorems 1.1', 3.1', 4.7', 5.4'

Proof of Theorem 1.1' (sketch in body §1.5). Part (a) is mechanical: ΔPV = −D · Δi/(1+i)² gives the asset-revaluation channel; concentrated asset holdings (top decile holds approximately 89% of equities by Federal Reserve, 2022) give the asymmetric incidence. The wage channel operates with the 6–18 month lag documented by Christiano, Eichenbaum, and Evans (2005). Part (b) is topological: the Cantillon network has finite diameter, money enters at specific nodes, prices do not adjust instantaneously, so propagation produces a real-purchasing-power gradient decreasing in d_C(i). Part (c) is by inspection: i_t = f(π_t, y_t) responds to scalar aggregates only; no distributional aggregate is in the rule's domain. For the Conditional Corollary, decompose the Gini dynamics under premises C1 (r > g) and C2 (φ < 1) as

dG/dt = (1 − φ) · [ω_1 ΔG_1(asymmetric) + ω_2 ΔG_2(Cantillon) + ω_3 ΔG_3(QE) + ω_4 ΔG_4(seigniorage)] + η(r − g)

where each ΔG_k > 0 by Propositions 1.1–1.4 in the body, all ω_k > 0, and η > 0 is the Piketty elasticity. Since (1 − φ) > 0 by C2 and (r − g) > 0 by C1, dG/dt > 0. The rate is bounded below by (1 − φ) · min_k(ω_k ΔG_k). □

Proof of Theorem 3.1' (sketch in body §3.6). The dynamics combine the three forcing terms (Piketty channel η_1(r−g)π_k, monetary channel (1−φ)η_2, stochastic channel η_3 σ²) and three counterforce terms (taxation −τ̄G, inheritance dilution −δG, entrepreneurial disruption −μ_d(G − G_L)) into

dG/dt = [η_1(r − g)π_k + (1 − φ)η_2 + η_3 σ²] − [(τ̄ + δ)G + μ_d(G − G_L)]

which is a first-order linear ODE dG/dt = A − BG with A = η_1(r − g)π_k + (1 − φ)η_2 + η_3 σ² + μ_d G_L and B = τ̄ + δ + μ_d. The closed-form solution is G(t) = Ḡ − (Ḡ − G_0)e^(−Bt) with Ḡ = A/B and half-life ln(2)/B. Ḡ < 1 for any B > 0; G(t) → Ḡ asymptotically, not in finite time. (μ_d as in the §A.1 glossary; bare μ elsewhere refers to the distribution mean.) □

Proof of Theorem 4.7' (full proof, building on body §4.4–§4.8). Step 1 (AGV transfers). Define t_i(θ) = E_{θ_{−i}}[Σ_{j≠i} v_j(f(θ_i, θ_{−i}))] − (1/(n−1)) Σ_{j≠i} E_{θ_{−j}}[Σ_{k≠j} v_k(f(θ_j, θ_{−j}))]. The first term is i's expected externality on others; the second term is a constant from i's perspective chosen for budget balance.

Step 2 (Ex-ante budget balance). Summing over i, E[Σ_i t_i(θ)] = E[Σ_i Σ_{j≠i} v_j(f(θ))] − E[Σ_i (1/(n−1)) Σ_{j≠i} Σ_{k≠j} v_k(f(θ))]. Both terms equal (n−1) · E[Σ_j v_j(f(θ))], so E[Σ_i t_i] = 0: ex-ante BB holds unconditionally. Ex-post BB would require independent types and onto allocation by Krishna and Maenner (2001); independence is partly engineered by defining types as residual Gini conditional on the global state Z_t = (g_t^W, p_t^W, i_t^W) per Pesaran (2006), but as flagged in §4.4 the gravity coupling and the AGV-transfer coupling endogenously reintroduce cross-sectional correlation. Ex-post BB is therefore a fragile refinement; ex-ante BB is the load-bearing claim.

Step 3 (BIC). The expected payoff to nation i from reporting θ̂_i is U_i(θ̂_i, θ_i) = E_{θ_{−i}}[v_i(f(θ̂_i, θ_{−i})) + Σ_{j≠i} v_j(f(θ̂_i, θ_{−i}))] + constant. The bracketed term is expected total social welfare when i reports θ̂_i; since f maximizes total welfare at the truth, U_i is maximized at θ̂_i = θ_i. Truthful reporting is a Bayesian Nash equilibrium. The BIC guarantee is conditional on the common-prior assumption flagged in §4.5; heterogeneous-prior environments are outside the BIC framework.

Step 4 (IIR). From §4.3: comparative-advantage gains plus monetary-stability premium exceed expected transfer contribution for all nations under reasonable parameter calibration.

Step 5 (Ex-ante Efficiency). By construction, f* maximizes Σ_i v_i(f(θ)) subject to feasibility; AGV transfers implement the efficient allocation in expectation. By the revelation principle, no other mechanism achieves higher expected total welfare while satisfying BIC + ex-ante BB.

On the absence of a DSIC strengthening. Myerson's (1981) one-dimensional payment identity under single-crossing on a Galor-Zeira credit-constraint channel would upgrade the mechanism to dominant-strategy incentive compatibility, but the EIMTS type space defined in §4.1 is five-dimensional and the Myerson machinery does not apply without a defensible scalar projection. Single-crossing on a defensible projection of θ_i is open work; the DSIC upgrade could be recovered under such a projection but the projection itself requires economic justification beyond this paper's scope. □

Proof of Theorem 5.4' (full proof). Steps 1–3 (excess-demand definition, upper-hemicontinuity via Berge's Maximum Theorem, Walras' Law) are standard. Step 4 (revised). The feasible set S = {x ∈ ℝ_+^n : A^ε(x) ≤ A*} is convex for ε > 1. Proof: A^ε(x) = 1 − Y_ede(x)/μ where Y_ede(x) = [Σ_i (x_i)^(1−ε)/n]^(1/(1−ε)). For ε > 1, (1−ε) < 0 so x_i^(1−ε) is convex; the sum is convex; the outer power 1/(1−ε) ∈ (−1, 0) is concave and decreasing on ℝ_+. The composition of a concave decreasing function with a convex function is concave (Boyd & Vandenberghe, 2004, §3.2.4). Therefore Y_ede is concave in x; A^ε is convex; the sub-level set S = {x : A^ε(x) ≤ A*} = {x : Y_ede(x) ≥ (1 − A*)μ} is the super-level set of a concave function and hence convex. Steps 5–6 (Kakutani fixed-point and equilibrium construction) proceed in the standard way with the Atkinson constraint as the binding object throughout. □

A.5 Additional Proofs

Proof of Theorem 2.4' (Conditional Trade-Structural Inequality). By the Stolper-Samuelson theorem, commodity-specializing periphery countries experience declining real wages as commodity prices fall (conditional on the factor-endowment divergence (i)). By the Prebisch-Singer mechanism, commodity prices decline secularly (mechanism in §2.3; conditional on the price-elasticity heterogeneity (ii)). By the gravity model, trade intensity is proportional to GDP and inversely proportional to multilateral resistance, creating a scale effect favoring large economies (conditional on the gravity-weighting pattern (iv)). By the core-periphery model, manufacturing concentration in core regions is stable and self-reinforcing (conditional on the agglomeration economies (iii)). By agglomeration effects, peripheral manufacturing costs remain higher, limiting diversification. Each mechanism is independently insufficient: counter-cases exist for each in isolation (Korea's industrial-policy escape from commodity dependence, the post-1990 declining-Gini cases in Brazil and Mexico, the recent commodity supercycles that temporarily reversed Prebisch-Singer terms-of-trade decline). The five mechanisms jointly imply the structural-bias pattern by conditional inference rather than by logical necessity. When all four joint conditions (i)–(iv) hold simultaneously, the peripheral country is trapped in a commodity-specialization equilibrium where commodity prices fall continuously, the value of exports declines, and the ability to afford imports of capital goods declines correspondingly. The wage gap ∂(w_peripheral − w_core)/∂t < 0 follows by joint conditional inference from the five mechanisms. No policy adjustment within the peripheral country alone can overcome the pattern unless the architecture itself is reformed. □

Proof of Theorems 5.5, 5.6', 5.7'' (Welfare Theorems). Theorem 5.5 (First Welfare) follows by contradiction: any feasible alternative satisfying market clearing, the Atkinson equality constraint A^ε(x) ≤ A*, and weakly improving all utilities while strictly improving at least one would have to violate the price-taking optimality at the equilibrium allocation, which contradicts utility maximization at p*. Theorem 5.6' (Existence-but-not-Operationality) follows from the standard second-welfare-theorem construction at the existence level: define new endowments ω_i' = (p* · χ_i − τ_i) where χ_i is the constrained Pareto optimal allocation; the resulting competitive equilibrium of the modified-endowment economy recovers χ. The operationality clause is not a corollary of the existence construction: the required Δω = ω_i' − ω_i must be financeable, and §1.4 (endogenous money) and §3.7 (existing-instrument calibration) jointly bound the financeable magnitude well below the required level for the contemporary global between-country distribution. The existence portion of the theorem is unconditional; the operationality clause is the explicit acknowledgment that the existence construction does not deliver an implementable redistribution mechanism from the architecture's monetary tools alone. Theorem 5.7'' (Conditional Kaldor-Hicks / Aggregate Welfare Improvement) holds when E(G_CE) − E(G*) ≥ Ē, where Ē is the magnitude of the externality reduction required to clear the aggregate compensation requirement under the constrained allocation; the inequality is conditional, not unconditional, because Ē is not estimated and only the sign dE/dG > 0 is invoked from the inequality-externality literature. The proof structure compares aggregate welfare under the constrained allocation, net of the externality, with aggregate welfare under the unconstrained competitive equilibrium baseline: when E(G_CE) − E(G*) ≥ Ē the aggregate (Kaldor-Hicks / potential-Pareto) improvement obtains. Conditional-on-Ē buys aggregate Kaldor-Hicks improvement rather than actual Pareto improvement at the household level, because Theorem 5.6' has already established that the compensating transfers that would convert Kaldor-Hicks to Pareto are not operational under current seigniorage and fiscal constraints. The conditioning is on the magnitude E(G_CE) − E(G*) ≥ Ē, not on the inequality-aversion parameter ε (which bounds the planner's aversion but not the empirical externality magnitude that must clear Ē). □

A.6 Stability Analysis: Theorem 5.1 and the Lyapunov Functions

Theorem 5.1 (Saddle-Path Stability). The equality-augmented system (5.5)–(5.6) exhibits saddle-path stability if and only if γ < γ̄, where

γ̄ = (σ_is β_y δ_e + κ σ_is α_π) / (κ σ_is η_e).

The characteristic polynomial of the Jacobian J is det(J − λI) = −λ³ − (σ_is β_y + δ_e γ)λ² − (κ σ_is α_π + σ_is β_y δ_e γ)λ − κ σ_is α_π δ_e γ + κ σ_is η_e γ = 0 (the λ here is the local eigenvalue variable in the characteristic-polynomial calculation — standard linear-algebra notation — and is not one of the four content-symbols disambiguated in the §A.1 glossary; the eigenvalue solution that determines the convergence rate of the closed-loop system is what becomes λ_conv in §7). By the Routh-Hurwitz conditions for asymptotic stability, the critical condition reduces to γ < γ̄ where γ̄ = (σ_is β_y δ_e + κ σ_is α_π) / (κ σ_is η_e), matching the statement above. For default calibration (κ = 0.3, σ_is = 1.0, α_π = 0.5, β_y = 0.5, η_e = 0.1, δ_e = 0.05), numerical computation yields γ̄ ≈ 1.70, confirming stability for all empirically plausible γ ≤ 0.40. For saddle-path stability under rational expectations, we require exactly one positive eigenvalue (jump variable π) and two negative eigenvalues, which holds for γ ∈ [0, γ̄]. □

For the Lyapunov analysis of Chapter 7, define V_2(z) = (1/2) z'Pz with P positive-definite. Differentiating along the system trajectory dz/dt = (A + BK)z with closed-loop matrix (A + BK), dV_2/dt = z'P(A + BK)z + z'(A + BK)'Pz/2 = z'[P(A + BK) + (A + BK)'P]z/2 = −z'Qz where Q = −(P(A + BK) + (A + BK)'P). For Q positive-definite (which holds when P solves the Lyapunov equation for the closed-loop system), dV_2/dt < 0, establishing asymptotic stability. The same construction with V_3(z) = (1/2)[(π−π*)² + α_1(y−y*)² + α_2(G−G*)² + α_3(e−e*)²] yields the Phase III stability proof. The convergence rate λ_conv = λ_min(Q)/λ_max(P) ≈ 0.20 per year under the standard calibration.

A.7 The Bridging Proposition (Gini ↔ Atkinson)

The Bridging Proposition stated in §5.6 admits exact computation for log-normal income distributions and conservative (containment-based) recovery for general distributions. For log-normal with log-variance σ², the Gini is G = 2Φ(σ/√2) − 1 (Φ standard normal CDF), and the Atkinson index is

A^ε = 1 − exp(−εσ²/2)

— the standard Atkinson-from-log-normal result. Both G and A^ε are monotone increasing in σ², establishing a bijection between G and A^ε via the shared σ² parameter.

Given G*, invert σ² = 2[Φ^{−1}((G*+1)/2)]² and compute A* = 1 − exp(−εσ²/2). For ε = 2 and G* = 0.30, the inversion yields σ ≈ 0.545, σ² ≈ 0.297, and A* = 1 − exp(−2 × 0.297 / 2) = 1 − exp(−0.297) ≈ 0.257. The Atkinson constraint A^ε(x) ≤ 0.257 corresponds to the Gini target G* = 0.30 for log-normal distributions at ε = 2.

For general distributions, the result is via Lorenz-domination containment. Atkinson's (1970) foundational theorem establishes that for distributions F and F' with the same mean, F Lorenz-dominates F' if and only if W(F) ≥ W(F') for every welfare function W satisfying the Pigou-Dalton transfer principle. Both Gini and Atkinson A^ε satisfy Pigou-Dalton; therefore on the Lorenz-comparable subset of distributions, the two orderings agree. For Lorenz-incomparable distributions (crossing Lorenz curves), Atkinson with ε > 1 is more sensitive to lower-tail transfers than Gini, making the Atkinson constraint strictly tighter in the relevant region. The containment S_A ⊆ S_G (where S_A = {x : A^ε(x) ≤ A*} and S_G = {x : G(x) ≤ G*}) therefore holds for appropriately chosen A*, and the equilibrium found under the Atkinson constraint satisfies the Gini constraint as a consequence.

This is what makes the Bridging Proposition the right honest framing. The Atkinson constraint is the formal binding object — convexity of S is what makes Theorem 5.4' go through. The Gini target G* survives as a reporting metric for policy audiences who prefer to think in Gini terms, with the conservative containment guaranteeing that any equilibrium satisfying the Atkinson constraint also satisfies the Gini constraint. The policy-implementation documents that present G* as the policy target are not technically wrong; they are simply about a derived metric rather than the formal object the equilibrium model commits to. □

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Appendix B: Data Sources and Replication

B.1 CGE Model Calibration

The computable general equilibrium model is calibrated to 2019 baseline data from the WIOD (World Input-Output Database) and COMTRADE (UN Commodity Trade Statistics). The model includes 25 countries/regions, 34 sectors, and was originally solved using GAMS (General Algebraic Modeling System); the implementation uses Python with scipy.optimize. The baseline equilibrium reproduces 2019 trade flows, domestic absorption patterns, and sectoral employment with a root-mean-square error of less than 2%.

Important: Bundled replication archive status (February 2026). The bundled replication archive GLOBAL_TRADE_EQUALITY_MODEL/10_Replication_Package/EIMTS_Replication_v3.zip corresponds to the pre-audit code that exhibits the documented bugs summarized in §6.6: hardcoded stability Jacobian, unenforced equality constraint in solve_equilibrium(), money-supply integration that computes but does not feed back, sign error in Gini dynamics rate effect, and broken welfare-aggregation routine. The CGE simulation results have been withheld from Chapter 6 rather than retained with disclaimer; the archive remains bundled with the paper as an open research artifact pending bug-fix work. Users running the archive will reproduce the prior simulation outputs but will not reproduce a constrained-equilibrium solver consistent with Theorem 5.4'. A corrected implementation is in development; the corrected archive will be released as EIMTS_Replication_v4.zip when the audit-identified issues are resolved. Researchers building on this work should use the corrected archive when available and should treat the prior archive's quantitative outputs as illustrative pre-fix outputs only.

B.2 Inequality Parameters

The Gini coefficient targets and inequality weights in the EIMTS simulations use a dual-source approach: World Inequality Database (WID) pre-tax national income shares for the primary calibration, and household-survey-based estimates from WIID (World Income Inequality Database) and LIS (Luxembourg Income Study) microdata for the sensitivity comparison. Country-specific Gini coefficients in the baseline range from 0.24 (Denmark) to 0.63 (South Africa). The Atkinson constraint in the corrected implementation will use ε = 2 with A* computed from the bridging proposition per Appendix A.7. The CGE code parameter gravity_gamma is renamed gravity_zeta in the corrected implementation to align with the notation glossary of Appendix A.1.

B.3 Trade Elasticities

Import demand elasticities are taken from Costinot and Rodríguez-Clare (2014) and average 1.5 across sectors. Export supply elasticities are set to 2.0 based on calibration to trade response functions. Bilateral distance data uses CEPII's GeoDist database. The gravity-equation equality elasticity ζ is treated as a free parameter pending estimation (see §5.3); an illustrative reference value ζ ≈ 0.15 with sensitivity bounds [0.10, 0.25] has been used in prior simulations, but the paper does not commit to this as a pilot estimate. The (1−σ) exponent in the corrected implementation matches the Anderson-van Wincoop form of eq. 2.27.

B.4 Monetary Policy Parameters

The standard Taylor Rule parameters (α = β = 0.5) are from Taylor (1993). The equality weight γ in the augmented rule is calibrated to range from 0 (baseline) to 0.40 (full equality targeting), with primary results at γ = 0.20. The equality-indexed money supply sensitivity λ_M (see Appendix A.1) is calibrated at 0.5 with sensitivity bounds [0.1, 1.5]. The composite-equality-index weights are w_1 = 0.50 (inverse Gini), w_2 = 0.30 (HDI), w_3 = 0.20 (GPI/GDP). See §5.2 for the GPI caveat and proposed conservative substitutes (Adjusted Net Savings, Inclusive Wealth Indicator). An implementation-ready calibration should substitute ANS or IWI for the GPI/GDP component to reduce the measurement-dispute surface area.

B.5 Simulation Scenarios

Main scenarios planned for the corrected implementation: (A) Baseline 2019 institutions through 2042; (B) Monetary-only EIMTS (γ = 0.20, ζ = 0, δ = 0); (C) Trade-only EIMTS (γ = 0, ζ = 0.20, δ = 0.20); (D) Full EIMTS (γ = 0.20, ζ = 0.20, δ = 0.20); (E) Aggressive EIMTS (γ = 0.40, ζ = 0.40, δ = 0.30). Sensitivity analyses are planned on each scenario across the WID-vs-survey calibration distinction and across ±50% perturbations of key elasticities. See §6.6 for the audit acknowledgment and the decision to withhold simulation results pending bug-fix work.

B.6 Replication Code

All code, data files, and documentation for the pre-audit implementation are available at the bundled replication archive (EIMTS_Replication_v3.zip). The corrected implementation (EIMTS_Replication_v4.zip) will be released when the audit-identified bugs are resolved; the public release will include an audit log mapping each fix to the corresponding audit finding. Estimated runtime for a single scenario is 2–4 hours on standard hardware (CPU-only); GPU acceleration of the equilibrium solver is planned in the corrected implementation. The paper does not report simulation results pending bug-fix work; researchers using the prior archive should treat its quantitative outputs as illustrative pre-fix outputs only, per the §6.6 audit acknowledgment.

Footnotes

1. Single-crossing for a defensible scalar projection of θ_i (most plausibly a residual-Gini projection holding (HDI, Y, T, r^d) at their joint conditional means) is an open problem. DSIC could be recovered under such a projection, but the projection itself requires economic justification beyond this paper's scope — specifically, it requires showing that the orthogonal components of θ_i do not bind in the mechanism's incentive constraints, which is itself a structural claim that single-crossing in the projected dimension does not imply. ↩