A Conditional Structural Diagnostic of the Luxembourgish Growth Model

Working Paper — Version 2 Classification: Decision-support document · Not an econometric forecast

Abstract

This paper decomposes the European Commission’s finding of declining Luxembourgish productivity into a Scissors Effect: the decoupling of an exogenous revenue dynamic, over which the jurisdiction is a price-taker, from an endogenous and indexed cost dynamic, over which it is a price-maker. In the central calibration the revenue blade reaches an index of 133 by 2040 against the cost blade’s 165 (2025 = 100), and the financial sector’s net margin compresses from 50% to 37.7% [M].

The fiscal consequence is contained on present assumptions, and it is contained by one variable. On the long-run compensation path the official ageing projections themselves assume, 2.9% a year, gross public debt reaches 56.5% of GDP by 2040 and crosses neither the EU reference value nor the 80% marker [M]. On the rate Luxembourg actually realised over the past decade, 3.33% a year, it reaches 70.8% and crosses the reference value in 2036 [M]. Both rates are sourced; the paper does not choose between them, because the distance between them is larger than any other driver in the model. Either path absorbs the single-tax-class reform, the family and childcare package, the defence commitment on the government’s own published trajectory, and a repaired treatment of the pension reserve. On the adverse configuration of every driver the ratio reaches 167.2% and crosses the 80% marker in 2033 [M].

The distance between those two paths is almost entirely one thing. A Shapley decomposition of the transition attributes 58.5% of it to the wage-indexation block alone, and the compensation growth rate on its own moves the 2040 debt ratio by 51.8 percentage points across the range the evidence supports [M]. The transition is 97% additive: there is no non-linear amplification to appeal to, and no doom loop.

That is the finding, and it is more useful than a debt number. The exogenous blade cannot be managed and does not need to be: it is not what determines the fiscal outcome. What determines the fiscal outcome is a domestic institutional arrangement, and the range of outcomes it spans is wider than everything else in the model combined. The sequence therefore stands: stabilise the cost base, capture the margin, then diversify.

Confidence in every parameter and finding is tagged ([H] hard / [M] medium / [A] assumption), and the parameters of greatest consequence have been calibrated against primary actuarial, central-bank, and statistical sources. The paper is a conditional diagnostic and not a forecast; the institutions cited are sources of published data and have neither reviewed nor endorsed it; nothing in it is financial advice.

Confidence apparatus

Every material claim, parameter, and projection carries a confidence tag:

• [H] — Hard. Drawn directly from a primary statistical or official source (SWD(2026) 216, CSSF, ESMA, IGSS, OECD, IMF, STATEC, BCL, Eurostat). Reproducible and externally verifiable.
• [M] — Medium. A defensible estimate built from primary data plus a transparent modelling step, or triangulated across sources.
• [A] — Assumption. An analyst judgement required to close the model. Directionally reasoned but not externally established.

The tags are not decoration. The paper’s methodological position is that the distribution of confidence across the argument matters as much as the argument itself: the diagnosis rests largely on [H] foundations, while the projections depend on parameters whose influence is bounded and reported explicitly, and whose hardening from [A] to [M] is documented in the companion Verification Report.

1. Introduction and Problem Statement

Luxembourg presents an analytical paradox. It is among the wealthiest economies in the European Union, with gross national income per head roughly sixty percentage points above the EU average [H], yet the European Commission’s 2026 country assessment (SWD(2026) 216) documents a sustained decline in measured labour productivity since 2010 — on the order of −0.5% per year [H] — at odds with the trajectory of most advanced economies. The prevailing diagnosis treats this as a productivity problem in the conventional sense: a weakening of the economy’s capacity to convert inputs into output.

This paper does not contest that diagnosis. It treats the Commission’s “declining productivity” finding as the primary hypothesis and subjects it to a structured sensitivity analysis of the drivers behind the measured decline. The distinction is consequential. If the decline reflects a genuine loss of internal operational efficiency, the appropriate policy response is the Commission’s: digitalisation, innovation diffusion, and R&D intensification. If, instead, a material share of the measured decline reflects an exogenous change in the terms of trade for Luxembourg’s dominant export sector — financial services — then efficiency-oriented policy addresses a symptom rather than the mechanism, and the binding constraints lie elsewhere.

The 2025 anchors. The analysis is anchored in the most recent official fiscal record. Every projection in this paper proceeds from an observed outturn rather than an assumption. The anchor set has been revised for this version; the earlier draft anchored nominal output on the 2024 outturn.

Table 1.1 — 2025 anchors

One property of the anchor set should be stated before a reader computes it independently and reaches a different number than the text: the debt ratio path is nearly invariant to the output anchor. The model is close to homogeneous of degree one in it, because revenue, spending, debt, the reserve and the property base are all expressed as ratios of it. Correcting the anchor barely moves the trajectory. What it corrects is every euro figure in the text, not the path.

The Conseil national des finances publiques (CNFP), Luxembourg’s independent fiscal institution, reports the 2025 general-government deficit of 2.0% of GDP and public debt of 26.5% of GDP recorded above [H]. The same evaluation records that 2025 net primary expenditure grew 8.1% against the 5.8% ceiling agreed with the EU Council [H] — an early, officially-documented instance of the expenditure dynamic this paper formalises. The detailed correspondence between that record and the model, and the more consequential divergence from the official medium-term sustainability analysis, are treated in Section 4.6. The forward path is anchored comparably: the IMF’s 2026 Article IV consultation independently projects gross debt above 39% of GDP by 2031 and locates the same financial-sector revenue assumption as the binding vulnerability (Section 4.6).

The tension the paper formalises is between two forces operating on different sides of the same ledger and governed by different actors:

1. An external, downward pressure on the price of financial services — global fee deflation, driven by the secular shift toward passive investment vehicles and intensifying fee competition. Luxembourg is a price-taker with respect to this force [H].
2. An internal, upward pressure on the cost of producing those services — labour costs locked to consumer-price inflation through automatic wage indexation, compounded by housing costs that the domestic policy framework has, on balance, subsidised on the demand side. Luxembourg is a price-maker with respect to this force [M].

We term the divergence of these forces the Scissors Effect. The remainder of the paper establishes its measurement foundations (Section 2), decomposes the structural risk it generates and the interaction among its domestic amplifiers (Section 3), projects the consequences under three calibrated scenarios to 2040 (Section 4), derives a policy sequence — rather than a policy list — whose ordering is itself a finding (Section 5), and records what changed between this version and its predecessor and why the change matters (Section 7).

A note on epistemic posture. This is a conditional diagnostic, not a prediction. Where the paper identifies a risk, it specifies the parameter on which that risk is contingent and the direction and approximate magnitude of the dependency. The objective is not to demonstrate that any particular outcome is inevitable — the analysis explicitly shows that the most favourable scenario retires public debt entirely — but to identify which levers are binding, which are secondary, and in what order they must be addressed for a stable path to remain reachable.

2. Methodological Framework

2.1 Purpose and epistemic status

This section establishes the analytical apparatus through which the paper interprets Luxembourg’s measured productivity record. The apparatus is deliberately modest in its claims. It does not seek to overturn the European Commission’s diagnosis of declining productivity (SWD(2026) 216); it seeks to decompose that diagnosis into its constituent drivers and to identify the conditions under which each interpretation of the measured decline holds. The exercise is therefore a conditional structural diagnostic rather than a forecast: its outputs are the sensitivities of the system to identifiable variables, accompanied by an explicit accounting of the confidence attaching to each.

The methodological contribution rests on a single observation. For an economy whose aggregate productivity statistic is dominated by one externally-oriented sector, that statistic inherits the measurement properties of the sector — including any ambiguity in how the sector’s real output is derived. In Luxembourg, financial and related services account for approximately one quarter of gross value added and of tax revenue [H]. The properties of the financial-services deflator are therefore not a technical footnote; they are a first-order determinant of what the national productivity series means.

2.2 The decomposition of output per hour

Let the nominal gross output of the fund-servicing sector be represented as the product of a price term and a volume term:

O_nom = P × Q

where P denotes the effective fee margin — revenue earned per unit of assets under management (AuM) — and Q denotes the volume of assets serviced. Dividing by hours worked, H, and expressing the measured real output per hour, O_h, as a function of the deflator Π applied to nominal output:

O_h ≡ (O_nom / Π) / H = (P · Q) / (Π · H)

This is an accounting identity, not a behavioural relation. Its analytical value lies in the separation of two economically distinct components of measured productivity: a price component, governed by the fee margin P that the global market pays for Luxembourg’s services, and a throughput component, governed by the operational efficiency Q/H with which those services are produced. The central interpretive question — whether a decline in O_h reflects deteriorating efficiency or compressing margins — reduces to the question of how the deflator Π treats movements in P.

In growth-rate form, denoting proportional rates of change by a dot:

Ȯ_h = Ṗ + Q̇ − Π̇ − Ḣ

The empirical inputs to this identity, in the paper’s central calibration, are summarised in Table 2.1. Each is grounded as indicated; none is a free assumption.

Table 2.1 — Calibration of the decomposition (central case)

The two structural facts that motivate the entire analysis are visible in this table and are individually [H] or well-grounded [M]: throughput is rising (Q̇ − Ḣ ≈ +4 pp/yr) while the price of the service is falling (Ṗ = −3 pp/yr). The sector is doing more work, more efficiently, for less revenue per unit. What the measured productivity series records depends entirely on whether the national accounts classify the falling fee margin as a price movement or as a volume movement.

2.3 The three deflation regimes

The treatment of Ṗ is determined by the choice of deflator Π. Three regimes are admissible under European System of Accounts (ESA 2010) practice, and they do not differ merely in magnitude: they differ in the sign of the resulting measured productivity trend.

Regime A — Fee-based price deflation. The deflator is the fee margin itself, Π̇_A = Ṗ. Margin compression is classified entirely as a price movement and removed from the volume measure. Measured real output then tracks the serviced volume Q:

Ȯ_h^A = Q̇ − Ḣ = 5.5 − 1.5 = +4.0 pp/yr    (productivity rising)

Regime B — Stock deflation by a general price index. Consistent with the Eurostat-recommended approach for financial services, the serviced asset stock is deflated by a broad price index and output volume moves with the real stock, Π̇_B = Π̇_gen applied to Q:

Ȯ_h^B = Q̇ − Π̇_gen − Ḣ = 5.5 − 2.0 − 1.5 = +2.0 pp/yr    (productivity rising)

Regime C — General-services deflation of fee revenue. Nominal fee revenue is deflated by a general services producer- or consumer-price proxy, Π̇_C = Π̇_gen applied to O_nom. Margin compression is retained within the volume measure:

Ȯ_h^C = Q̇ + Ṗ − Π̇_gen − Ḣ = 5.5 − 3.0 − 2.0 − 1.5 = −1.0 pp/yr    (productivity falling)

2.4 The price-measurement contingency as a bounded magnitude

The paper’s treatment of these regimes is its principal methodological commitment, and it is deliberately conditional. The difference between a volume-consistent regime and Regime C is not presented as a discovery that the measured decline is “really” a price illusion; it is presented as a bounded magnitude whose realisation depends on a documented statistical choice:

Δ_illusion = Ȯ_h^A − Ȯ_h^C = Π̇_gen − Ṗ = +5.0 pp/yr

Δ′_illusion = Ȯ_h^B − Ȯ_h^C = −Ṗ = +3.0 pp/yr

The price-measurement contingency is therefore quantified at 3.0 to 5.0 percentage points per year of measured output-per-hour growth [M] — the interval bounded by the choice between Regimes B and A on the volume-consistent side. This is the magnitude by which the choice of deflator, alone, displaces the measured productivity trend.

The analytical consequence is symmetric and is stated without rhetorical preference for either outcome:

• If Regime C is in force, the Commission’s observed financial-sector decline (output per hour falling in most years since 2016 [H]) is fully consistent with rising true throughput; the measured decline would be substantially a price-measurement artefact.
• If Regime A or B is in force, the same observed decline indicates a genuine throughput component, because those regimes already remove margin compression from the volume measure; under Regime B a residual decline of part of the observed magnitude would remain attributable to efficiency.

It is important to record that even Regime B — the approach Eurostat methodology favours for financial services — implies rising measured productivity (+2.0 pp/yr) on the central calibration, which itself does not reconcile with an observed decline. Where the observed series declines under a volume-consistent regime, the decline carries genuine efficiency information. The paper does not assert that the entire measured decline is artefactual; it asserts that the artefact component is bounded at 3.0–5.0 pp/yr and that the residual is a genuine efficiency signal whose size is determined by the deflator in use.

2.5 Standard practice and residual uncertainty

The Eurostat Handbook on Price and Volume Measures in National Accounts (ESA 2010 edition) constrains, without fully resolving, the regime question. For financial services the handbook recommends the stock-deflation and output-indicator methods — Regime B-type approaches and quantity-based equivalents — in preference to deflation of nominal output by a general services price index. It further deprecates the use of broad price indices for output deflation and stresses derivation “at the most detailed level possible,” with quality adjustment, using producer-price indices, charge-out rates, or unit values where available.

Two considerations prevent this from closing the contingency:

1. The explicitly-charged fund-management activities that dominate the Luxembourg aggregate (NACE K64.3, K66.3) are distinct from FISIM and are deflated separately. Where no specific fund-fee price index exists — and none is published for Luxembourg — the compiler’s fallback may approximate Regime C (a broad service deflator) or Regime A/B (an output indicator). The fallback is precisely where the contingency lives.
2. The treatment is not documented publicly at the granularity required to determine the regime with certainty.

Accordingly, the contingency is narrowed but not eliminated: Eurostat methodology renders a pure Regime C outcome unlikely, which places the most defensible interpretation toward the volume-consistent regimes and toward a blended reading of the measured decline — part genuine efficiency, part price artefact, with the artefact share bounded as above. The single remaining step required to convert this parameter from [M] to [H] — a direct methodological inquiry to STATEC regarding the K64.3 / K66.3 deflator — is documented separately and does not block the analysis, because, as the following sections establish, the paper’s structural conclusions rest on the cost side of the model, which is independent of the deflator question.

2.6 The modelling object and its limits

The quantitative results in Section 3 and in the scenario analysis derive from a transparent recurrence-relation model of the Luxembourg economy to 2040, calibrated to the anchors above and driven by the interaction of a revenue path (volume growth net of fee deflation) and a cost path (compensation growth under wage indexation). The model is a strategic decision-support instrument and not an econometric forecast. It incorporates only those behavioural feedbacks that are explicitly specified: a housing-wealth channel from the residential price index to non-financial value added; a sovereign-financing channel in which a debt ratio above a stated threshold raises the marginal cost of debt; and the transmission of wage indexation simultaneously to private-sector compensation and to indexed components of public expenditure. Its outputs are decision-relevant magnitudes and orderings rather than point predictions. Every parameter carrying an [A] tag is subjected to the sensitivity analysis reported in Section 3.2, and the parameters of greatest consequence have been hardened to [M] against primary sources, as documented in the Verification Report.

3. The Drivers of Structural Risk

3.1 The Scissors Effect: a decoupling of governance

The structural risk to the Luxembourgish model originates not in either of its two principal price dynamics considered alone, but in the fact that they are governed independently — that the price the economy receives for its dominant export and the price it pays to produce that export respond to different signals and are set by different actors. We term this decoupling the Scissors Effect. Its two blades are characterised as follows.

The revenue blade is exogenous. The fee margin earned on financial services is set in global markets. Margin compression is a secular, structural phenomenon driven by the substitution toward passive investment vehicles, by fee-transparency regulation, and by the bargaining power of institutional allocators. Luxembourg possesses no domestic instrument capable of arresting it; with respect to this force the jurisdiction is a price-taker [H]. In the central calibration, the sector’s domestic value capture rises to an index of 133 by 2040 (2025 = 100) [M] — growth, but growth attenuated by the downward pressure on P.

The cost blade is endogenous. Compensation in the sector grows under automatic wage indexation, which ties it to consumer prices rather than to the price of what the sector sells. Over 2015 to 2025 compensation per employee in financial and insurance activities grew at 2.94% a year [H, Eurostat nama_10_lp_ulc, vintage 2026-09-08]. The 2024 Ageing Report’s long-run framework, in which real wages track labour productivity at 0.9% a year against inflation of 2.0%, implies approximately 2.9% a year in nominal terms [M, via IGSS Cahier statistique n°18, July 2024]. The two agree, and the central calibration uses 2.9%. On that path the cost blade reaches an index of 165 by 2040 against the revenue blade’s 133.

A correction the paper makes explicitly, because it contradicts an intuition the earlier draft relied on: the financial sector does not pay above the index. Over the same decade its compensation per employee grew more slowly than the whole economy’s, 2.94% against 3.33% [H]. An earlier calibration applied a single rate of 3.6% to both the sector and the public finances, above each of the two series it stood for.

Table 3.1 — The two blades, 2040 (2025 = 100)

V1 column reproduced from the June 2026 version for comparison; V2 column is this paper’s central calibration [M].

The two blades therefore diverge across the projection horizon — the cost index reaching 165 against the revenue index’s 133 — and that divergence is the quantitative statement of the squeeze. The economically salient feature is not the level of either blade but their decoupling: a jurisdiction whose flagship industry faces exogenous, uncontrollable revenue deflation while its principal cost is indexed to a domestic price level over which the sector has no influence will experience a structural compression of net margin irrespective of its operational performance. In the central calibration this compression carries the sector’s net margin from approximately 50% in 2025 to 37.7% by 2040 [M]. This figure is the scissors’ signature, and it is independent of the Section 2 deflator contingency: it is a statement about nominal revenues and nominal costs, not about how real output is measured.

3.2 The domestic amplifiers and the structure of sensitivity

Several domestic variables govern the cost blade and the fiscal capacity to absorb its consequences, of which the most consequential are wage indexation, the fiscal–pension trajectory, and housing. The relevant policy question is not which of these is “most important” in the abstract, but which, when moved from its central to its adverse calibration, most displaces the long-run fiscal position — and whether that displacement crosses a threshold of concern.

The sensitivity analysis isolates each driver in turn, moving it alone from the central case to the adverse end of the range its own source supports, holding the remainder at their central values, and records the effect on the 2040 gross-debt-to-GDP ratio. Because the adverse settings are calibrated against primary sources rather than chosen, the resulting ranking is not an artefact of arbitrary stress assumptions; it is a direct consequence of disciplined calibration. Drivers for which no range could be sourced are not varied and therefore do not appear. The ranking is reported in Table 3.2.

Table 3.2 — One-at-a-time sensitivity of 2040 debt/GDP (central baseline = 56.5%)

One driver is worth more than every other driver in the table combined. That is the paper’s result, and it is a statement about where policy has purchase rather than about how bad the outcome will be. The two rates that bracket the central case are not a favourable and an adverse assumption; they are two published measurements of the same quantity, one forward-looking and official, one realised and statistical, and the gap between them is 14.3 points of 2040 debt.

How indexation reaches the debt. It reaches it through public spending and through nothing else. The sector’s own wage bill affects the margin and never touches the fiscal balance. Public primary spending grows at the indexed share of that expenditure multiplied by the wage rate, plus the deflator on the remainder. The indexed share is the transmission coefficient, and no institution publishes one. Built from the expenditure composition and the indexation law it is 58.5% on the strict legal core, public pay and cash social benefits, and 75.5% on the Conseil National des Finances Publiques’ own list of lines it describes as directly influenced by the index, which adds benefits in kind and intermediate consumption [M, Eurostat gov_10a_main 2024, vintage 2026-07-21; CNFP, Évaluation des finances publiques, March 2024, p.26]. The central calibration uses 0.67, the midpoint that treats the two contested categories at half weight. Neither bound supports the 0.80 an earlier calibration assumed.

A consequence worth stating: for indexation to be neutral for the public finances, the tax base would have to be indexed to wages as fully as the spending base is. It is not, and the reform before the Chamber makes it less so, by adjusting tax brackets to inflation automatically and removing the fiscal drag that partly offset indexation on the revenue side.

Two features of this ranking are central to the diagnostic and warrant deliberate emphasis, because each revises a common prior.

First, the dominant fiscal risk is domestically controllable. Compensation growth transmitted through the public channel occupies the first rank by a wide margin, carrying many times the fiscal impact of exogenous fee deflation — the variable to which external commentary most frequently attributes Luxembourg’s structural risk. This is the analytically consequential inversion of the conventional reading. The force that most threatens long-run sustainability is not a global market dynamic over which the jurisdiction has no control, but a domestic institutional arrangement over which it retains full policy authority. Fee deflation, being gradual, exogenous, and partially offsettable through the capability-capture strategy discussed in the policy section, ranks well below it.

This finding receives independent corroboration from the actuarial record. The International Monetary Fund’s 2025 Technical Assistance assessment attributes approximately 81% of the upward revision in the projected pay-as-you-go contribution rate (which advanced the pension system’s cash-deficit onset from 2028 to 2026) to a single revised assumption: lower employment growth [H]. The pension trajectory and the margin squeeze thus share a common mechanism. Employment growth is itself a function of the cost competitiveness that the scissors compresses; the fiscal–pension clock and the revenue–cost divergence are not independent risks sitting side by side but two expressions of the same underlying dynamic.

Second, housing is not a large fiscal amplifier. An earlier, pre-validation calibration assigned housing a materially larger fiscal impact and a correspondingly higher rank. Calibration against the Banque centrale du Luxembourg’s overvaluation series revised this. The BCL records that the residual house-price overvaluation had fallen to approximately 13.4% by end-2024 [H], from a euro-area-high peak of 51.3% in 2021Q2 [H], the intervening correction having been realised — a price trough of −14.5% year-on-year in 2023Q4 [H] — in what the IMF characterises as a substantial but orderly correction. Because the overvaluation that would fuel a large forward correction has already substantially unwound, the plausible forward correction is bounded, and the housing-stress impact falls to +2.7 pp — far below the 80% threshold in isolation. Housing’s claim on policy priority, developed in the policy section, therefore rests on a political-economy argument concerning binding constraints on labour retention and cost competitiveness, and not on it constituting a large fiscal lever. The demotion of housing is a direct product of the confidence apparatus: housing-correction depth was the [A] parameter whose disciplining against [H] data most changed the analytical picture.

3.3 The interaction between the amplifiers

The housing and fiscal–pension blocks interact, and the interaction is super-additive. The magnitude, however, must be stated with precision, because the policy implication depends on it and because an exaggerated characterisation would not survive scrutiny.

The interaction operates through an identifiable channel. A housing correction widens the fiscal deficit — through reduced property-related revenue and the fiscal cost of transition — which raises the debt path, which can cross the threshold at which the sovereign-financing channel raises the marginal cost of debt, which widens the deficit further. The mechanism is real and the direction is unambiguous. Its magnitude is small, and it is measured rather than asserted: across the full driver set, the Shapley decomposition reported in Section 4.4 puts the entire interaction term at 3.8 percentage points against a transition of 110.7 points, so that the transition is 97% additive [M].

The paper deliberately declines the stronger characterisation. The interaction is a compounder of risk, not a self-reinforcing collapse mechanism: a non-linearity of this size does not constitute, and should not be described as, a “doom-loop.” The policy-relevant inference is correspondingly precise and more useful than a collapse narrative would be. Because the blocks compound — even modestly — the sequencing of reform carries a cost of error: addressing housing and fiscal policy in isolation understates their joint fiscal cost by the super-additive excess, and a housing correction initiated before the mortgage book has been de-risked transmits into the financial system through precisely the channel the interaction describes. The implication is therefore not urgency born of catastrophe, but ordering born of compounding — a point the policy section develops into a sequencing constraint.

3.4 Synthesis: the structure of the diagnostic

The drivers analysis yields a structured picture rather than a single headline. The Scissors Effect (3.1) establishes the mechanism: a structural compression of net margin arising from the decoupling of an exogenous revenue dynamic from an endogenous, indexed cost dynamic, quantified independently of the deflator contingency. The sensitivity ranking (3.2) establishes the hierarchy: the dominant fiscal risk is a single domestically-controllable lever — the compensation growth rate transmitted through indexed public spending — whose effect on the 2040 ratio exceeds that of every other driver in the model combined, and exceeds by a wide margin that of exogenous fee deflation, with the pension trajectory shown by the IMF to share the scissors’ employment-growth mechanism. The interaction analysis (3.3) establishes the qualification: the domestic amplifiers compound only modestly, so the analysis is an ordered set of largely additive risks in which sequencing matters for the cost of error rather than for the avoidance of a collapse.

The diagnostic is conditional throughout. It does not assert that any particular fiscal outcome is inevitable; the scenario analysis demonstrates that the favourable calibration retires public debt entirely. It asserts, rather, that the variables determining which outcome obtains are identifiable, that they are predominantly domestic and therefore within the reach of policy, and that the order in which they are addressed is itself a determinant of feasibility. It is to the quantification of those outcomes, under explicitly bounded scenarios, that the analysis next turns.

4. Scenario Simulations: Projecting the 2040 Horizon

4.1 Status of the simulations

The structural drivers established in Section 3 are applied to the project’s recurrence-relation model and projected from 2025 to 2040. These are simulations, not econometric forecasts: they map the arithmetic consequences of specified parameter configurations, and their value lies in the orderings and thresholds they expose rather than in any claim to predict realised outcomes [stated convention]. The three scenarios correspond to internally consistent calibrations of the driver set, each satisfying the same accounting identities and differing only in the configuration of the levers analysed above. No probabilities are assigned; the scenarios bound a plausible range and isolate the transitions between regimes.

Method note. The tails are not chosen. Each is built by setting every driver to the end of the range its own source supports, starting from the central case. Nothing in either tail is a hand-picked value, and every value in them carries a citation by construction. Drivers for which no range could be sourced are not varied, so the tails understate rather than overstate the spread.

4.2 Scenario definitions

Scenario O — the net-asset-accumulation case. This configuration assumes the successful stabilisation of the cost base: a decoupling of wage indexation from high-volatility inflation, a stabilised pension trajectory, and sustained value capture in the financial sector. Its defining feature is not the absence of debt but the accumulation of a net financial-asset position. The configuration generates sustained primary surpluses that retire gross public debt within the projection horizon — the ratio stands at 0% of output at 2040 [M] — and accumulate a net financial-asset position thereafter. This is the analytically meaningful characterisation of the favourable case — a sovereign net-creditor position — and it is reported not because it is probable (it requires primary surpluses few governments sustain) but because it establishes that the trajectory is conditional rather than determined.

Scenario C — the managed-erosion case. This configuration continues current structural trends: wage indexation remains tied to consumer inflation, the pension cash deficit follows the IGSS/IMF-projected path, and global fee-margin compression continues at observed rates. It produces a steady increase in the debt ratio to 56.5% of GDP by 2040 [M], with the scissors operating as modelled and the pension reserve and sovereign rating remaining intact. It is consistent with the European Commission’s qualitative diagnosis while attributing the mechanism to margin compression rather than to efficiency loss.

Scenario P — the structural-transition case. This configuration sets the driver set to its adverse calibration: every driver at the adverse end of the range its own source supports, including compensation growth and the indexed spending share, compounded by the housing, defence and fee blocks and by the sovereign-financing channel that activates above the debt threshold. It is not merely a worse instance of the Central case but a transition to a distinct equilibrium — high-debt, low-margin, and high-volatility. Gross debt reaches 167.2% of GDP [M], the financial-sector net margin falls to 9%, and the pension reserve is drawn down materially, to €24bn at the horizon.

4.3 Comparative projections

Table 4.1 — Comparative projections, 2040 horizon

The two central rows. The central column is calculated on a public-sector compensation growth rate of 2.9% a year, which is what the 2024 Ageing Report’s long-run framework implies and the only official assumption that reaches a 2040 horizon [M]. The alternative is 3.33%, the rate the whole economy realised over 2015 to 2025 [H, Eurostat]. On that rate the central configuration’s 2040 gross debt ratio is 70.8% rather than 56.5%, and the reference value is crossed in 2036 [M]. Both rates are defensible and the paper reports both, because the choice between them moves the 2040 ratio by 14.3 points and decides whether the central path crosses the reference value at all. It is the largest single judgement in the model and it is a judgement between two sourced numbers, not between evidence and assumption.

The presentation of the financial-sector margin warrants precision. In Scenario P the net margin does not vanish; it falls to 9%, the limiting condition being that the cost blade converges on the revenue blade. Economically, this is the region in which domestic operating and compensation costs consume very nearly the entirety of the sector’s gross fee revenue, and the financial sector’s net fiscal contribution is sharply reduced — a structural condition, not a rounding artefact [M].

4.4 Variance decomposition of the structural transition

The central analytical question raised by Table 4.1 is what drives the transition from the Central debt level (56.5% of GDP) to the Adverse level (167.2%) — a movement of 110.7 percentage points. It is essential to attribute this transition correctly, because a misattribution to interaction effects would materially misrepresent the nature of the risk.

The transition is decomposed by a Shapley decomposition over eight driver blocks, computed across all 256 coalitions, which distributes the full transition — including all interaction effects — across the drivers symmetrically, so that each driver receives its marginal contribution averaged over every possible ordering of the drivers. The Shapley values are conserved to the transition exactly and constitute the formally correct attribution of each driver’s share.

Table 4.2 — Variance decomposition of the 110.7 pp transition (Central → Adverse)

All values [M]; Shapley values verified to sum to the full transition.

The transition from central to adverse is 110.7 points and is 97% additive; the interaction term is 3.8 points. The refusal the earlier version made stands, and is stronger here: there is no doom loop. An interaction term of 3.8 points on a transition of 110.7 points is not an amplification mechanism, and describing it as one would be a quantitative error of the first order. The level is built, additively, from the stacking of domestic drivers.

Two findings follow, and two caveats must travel with the table or it will be misread.

First, a single domestically-controllable lever dominates. On the Shapley attribution, the Wage indexation block accounts for 58.5% of the entire transition [M] — more than every other block combined. The macroeconomic and revenue environment, the exogenous backdrop of AuM growth, sector efficiency, and the deflator, contributes 15.1%. The structural transition is, in its arithmetic, a story about one domestic policy variable.

Second, the shares are not comparable with the earlier version’s. That decomposition varied a different and larger set of parameters, several of them without sources. Two blocks contribute 0 pp here because the tails vary only drivers with a sourced range, and no range could be established for the servicing-leakage or job-formation parameters. Their contribution is a consequence of the sourcing discipline and not a measurement of their unimportance.

The wage-indexation block now contains three parameters — the sector rate, the public rate and the indexed share — because all three are the same transmission channel and separating them would attribute a single mechanism to three different places.

4.5 Critical thresholds as decision-support markers

The scenario set localises two thresholds of direct relevance to policy. Both are presented as decision-support markers — points at which the system’s sensitivity to further shocks rises steeply — and not as mechanical cliffs.

The debt/GDP markers. On the central configuration the debt ratio reaches 56.5% by 2040 and crosses neither the EU reference value of 60% nor the 80% sustainability marker [M]. That result is conditional on one number, and the paper states the condition rather than burying it. The central calibration uses a public compensation growth rate of 2.9% a year, the 2024 Ageing Report’s long-run assumption. On the 3.33% the whole economy realised over 2015 to 2025 the same model reaches 70.8% and crosses the reference value in 2036 [M]. Around the central value the gearing is approximately 3.3 percentage points of 2040 debt for each tenth of a percentage point of compensation growth.

No other driver in the model comes close to that. From the central baseline only one single driver carries the ratio across the 80% marker: compensation growth through the public channel, at +51.8 pp (Table 3.2). No other single driver reaches it — the next largest, AuM growth, contributes +15.7 pp. The reader who wants a single sentence should take this one: whether Luxembourg’s debt ratio stays inside the European reference value to 2040 is decided by the wage path, and by very little else.

On the adverse configuration of every driver the ratio crosses 60% in 2031 and 80% in 2033, reaching 167.2% [M]. The favourable configuration retires the debt entirely.

The paper’s earlier version reported a central path crossing 60% in 2038. That path rested on a compensation growth rate and an indexed spending share that carried no citation and that were, on examination, above every realised series and every official assumption. The correction is reported here rather than absorbed, because the direction is inconvenient: the diagnostic’s own central case is now more benign than the version that preceded it, while the mechanism it describes is unchanged and the range it spans is wider.

The 80% level remains a modelling convention for a sustainability inflection, not a legal or universal limit, and the 60% is a reference value rather than a cliff [stated convention].

Pension reserve depletion. The reserve-depletion date is the marker that most sharply distinguishes a managed path from a structural transition. Anchored to IGSS, OECD, and IMF actuarial projections [H], the pre-reform reserve is projected to deplete between 2045 and 2047; the 2025 reform (the contribution-rate increase to 25.5% from 2026) extends this to approximately 2050. In the model, the reserve cushions the pension cash deficit through 2040 in all three configurations — it stands at €34bn, €31bn and €24bn respectively at the horizon [M] — and is drawn down furthest in the Adverse configuration. The model’s behaviour thus sits inside the official actuarial envelope: the reserve is not depleted within the projection horizon under any of the three paths, but the pension cash deficit it absorbs grows throughout, and the post-2040 acceleration the OECD identifies lies beyond the horizon of this analysis and constitutes a distinct, longer-dated risk. This version also repairs an identity under which drawing the reserve retired gross debt, an identity that had made every pension parameter in the earlier model inert (Section 7).

4.6 External corroboration: the CNFP outturn and the IMF Article IV baseline

The scenario analysis can be situated against two independent official references: the May 2026 evaluation by the Conseil national des finances publiques (CNFP), Luxembourg’s independent fiscal institution, which anchors the 2025 outturn; and the June 2026 IMF Article IV consultation (Country Report No. 26/159), which anchors the forward path. One distinction governs the use of both. The IMF assesses Luxembourg’s debt as sustainable, at low overall risk of sovereign stress, with fiscal space that is ample though narrowing [H, Annex IV]. This paper neither disputes that nor advances a sovereign-stress thesis; its subject is structural margin compression and competitiveness, on which the Fund’s diagnosis and this paper’s coincide. What follows is corroboration of the debt trajectory and its drivers — not of any default-risk interpretation. The CNFP evaluation is short-horizon and compliance-oriented and does not itself model the medium term, decompose drivers, or address productivity; the IMF supplies the medium-term projection the CNFP does not.

Corroboration of the mechanism. Beyond confirming the initial conditions noted in Section 1 (2025 debt 26.5% of GDP; deficit 2.0%), the CNFP documents the cost blade directly. Net primary expenditure grew 8.1% in 2025 against the 5.8% ceiling — an annual deviation of 1.03% of GDP, exceeding the 0.3% reference threshold [H] — and the contemporaneous expenditure dynamic is attributed to compensation of employees, current transfers, and intermediate consumption rather than to investment [H]. The institution further records that the restraint the plan requires (averaging 4.9% over 2025–2028) lies 1.7 percentage points below the historical norm of 6.6% (1996–2023) [H]. In the language of expenditure rules, this is the proposition the paper advances in the language of margin compression: the cost base is expanding faster than the sustainable path, the drivers are indexed and domestic, and stabilising them requires a departure from the historical trajectory.

The leading-indicator observation. The official debt-sustainability analysis underpinning the plan did not project the deficit to reach 2.1% of GDP until 2038 [H]. The 2025 outturn of 2.0% reached that level thirteen years early. The deterioration the official baseline distributed across the projection horizon materialised in the first year of the plan — consistent with this paper’s characterisation of the central path as an erosion already under way rather than a distant risk.

The forward path: the IMF as the middle baseline. The decisive evidence is that a second independent institution projects the same direction of travel. The IMF’s 2026 Article IV baseline has gross debt rising from 26.5% of GDP in 2025 to above 39% by 2031 under current policies, a rise it attributes principally to current expenditure [H, ¶16]. The three baselines stand in a revealing order.

Table 4.3 — Three baselines for gross debt / GDP (%)

Authorities’ DSA: PBSN (October 2024), Table 2.3 [H]. IMF: Country Report 26/159, Table 1 and Annex IV [H]; the DSA horizon ends at 2035. Central scenario: this paper’s recurrence model [M], every value computed rather than transcribed.

Two features bear on the argument. First, the authorities’ own analysis is the outlier: it holds debt near 26% through the late 2030s, while both the Fund and this paper project a rise. Second, and more consequentially, the Fund locates the source of the gap in the same terms this paper does — its baseline contrasts with the authorities’ plan because the latter rests on “more optimistic revenue assumptions” [H, ¶15].

Two features bear on the argument. First, the authorities’ own analysis is the outlier: it holds debt near 26% through the late 2030s, while both the IMF and this paper project a substantial rise by 2031. The benign official path is therefore not corroborated by the Fund; the adverse path is. Second, and more consequentially, the IMF locates the source of the gap in the same terms this paper does. Staff state that their baseline contrasts with the authorities’ medium-term plan because the latter rests on “more optimistic revenue assumptions,” and that “the financial sector’s outperformance and a rising labor income share are expected to wane, with tax buoyancy gradually reverting to historical averages” [H, ¶15]. The authorities, in turn, remain confident in sustainability “underpinned by sustained revenue growth from the financial sector’s strong performance” [H, ¶21]. The disagreement is thus precisely located: the official baseline assumes the financial-sector revenue base holds; the IMF and this paper both identify that assumption as the binding vulnerability. The scissors of Section 3 is the structural account of why it does not hold.

The paper’s central scenario and the Fund’s baseline agree in direction and in mechanism, and differ in level by a margin of a few percentage points through the 2030s, against an authorities’ path that holds debt near 25.5% throughout. The residual difference rests on a single assumption: the Fund treats the erosion of financial-sector tax buoyancy as cyclical and mean-reverting, whereas this paper treats it as structural — the secular fee-margin compression of Section 3, which does not mean-revert. The widening gap in the 2030s is therefore not a disagreement over the present, on which the two baselines agree, but the quantitative expression of the central claim: that the revenue blade’s decline is a level shift in the global price of intermediation, not a phase of a cycle.

The official DSA assumes, by construction, that revenues evolve in line with nominal GDP growth [H]; the IMF rejects that assumption qualitatively, and this paper quantifies its failure. If the financial-sector fee margin compresses as modelled (Section 3), financial-sector tax revenue grows more slowly than nominal GDP, and the revenue side of the official baseline is overstated.

The magnitude is quantifiable within the model. In the central calibration, financial-sector domestic value capture grows more slowly than nominal output, the revenue blade reaching an index of only 133 by 2040 against a cost blade of 165 [M]. Financial-sector tax revenue, approximately one quarter of the total [H], therefore does not hold its share of output: under the scissors that share declines year by year, against the constant share the official “revenues track nominal GDP” assumption implies [M]. Financial-sector tax revenue falls from 11.8% of output in 2025 to 9% by 2040, against the constant share the official assumption implies. The annual shortfall widens to 2.8 percentage points of output by 2040 and cumulates to approximately 23 percentage points over 2025–2040 [M]. That cumulative shortfall, carried onto the debt stock, accounts for the larger part of the divergence from the authorities’ path; the remainder is attributable to the pension trajectory (Section 4.5) and to the structurally wider primary deficit the 2025 outturn already reveals.

What is like-for-like and what is not. The Fund’s June 2026 baseline already carries the single-tax-class reform and the family and childcare package [H, CR 26/159 ¶15 and Annex V]. Those commitments are therefore not a divergence between this paper and the Fund; they are a divergence between both and the authorities’ plan, which the Fund notes “does not yet account for the impact of income tax reform and additional support measures” [H, ¶15]. The near-term difference runs the other way and should be stated: this model’s projected balance is more adverse than every official estimate from 2027 onward, and no adjustment consistent with its own sources closes that gap. The first projected year is anchored on the official figure of −1.2% of output [H, EDP notification of April 2026; European Commission Spring 2026 forecast]; the years after it are the model’s own.

The contribution of this paper is therefore not to contradict the official sustainability analysis on its own terms — within its assumptions it is internally coherent — but to identify and stress-test the revenue assumption it holds fixed, and to show that the 2025 expenditure overshoot the CNFP documents is the first observable instance of the mechanism by which the benign baseline ceases to hold.

Scope discipline. The CNFP evaluation does not address the pension system’s actuarial trajectory, the housing market, or sectoral productivity; the corroboration drawn from it is confined to the 2025 fiscal outturn and the expenditure mechanism. The IMF baseline corroborates the trajectory and drivers through 2031 and, on its own DSA horizon, to 2035; the 2035–2040 segment of the central path is more adverse than any external baseline and rests on the structural-continuation assumption set out above, which is this paper’s to defend. The pension and housing parameters are hardened against the IGSS/OECD/IMF and BCL sources documented in the Verification Report.

5. Conclusion and Policy Sequencing

5.1 The structural diagnosis

The analysis supports a diagnosis that is structural rather than cyclical. Luxembourg’s measured productivity challenge is, on the evidence assembled here, substantially a problem of decoupled prices rather than of deteriorating internal efficiency — a conclusion whose precise weight is contingent on the deflator methodology examined in Section 2 (the price-measurement contingency, bounded at 3.0–5.0 pp/yr of measured output-per-hour growth) but whose direction is robust to that contingency. The financial sector services a growing volume of assets at a falling fee margin while paying a domestically-indexed cost base; the resulting compression of net margin — from 50% to 37.7% of gross revenue in the central case, and to 9% in the adverse case — is the structural signature of the model.

The decomposition in Section 4.4 establishes the second principal finding: the transition to a high-debt regime is predominantly additive (97%) and is dominated by a single domestically-controllable lever — the Wage indexation block — which alone accounts for 58.5% of the movement, more than every other block combined. This is the analytically consequential result. The force that most threatens long-run sustainability is not an exogenous market dynamic over which the jurisdiction has no authority, but a domestic institutional arrangement over which it retains full policy control. The risk is, in the precise sense established by the variance decomposition, endogenous and controllable.

It should be stated plainly that on the central configuration the trajectory is not alarming: the ratio reaches 56.5% by 2040 and crosses neither the reference value nor the 80% marker [M]. The finding is not a debt number. It is that the range of outcomes the model spans is set almost entirely by one lever.

5.2 Why sequence is a structural property, not a preference

The ordering follows from the decomposition rather than from the level of the debt. Stabilising the cost base is first not because the trajectory is alarming — on the central configuration it is not — but because that single lever spans a wider range of outcomes than every other driver combined, and because it is the one the jurisdiction controls outright. Capturing the margin is second because it acts on a blade that is exogenous in price and can only be addressed in volume. Diversification is third and is not a substitute for either: on the calibration here it moves the 2040 debt ratio by 0 points, and representing it as an answer to the fiscal question would misstate its quantified contribution.

Two further properties of the model reinforce the ordering. First, certain reforms are preconditions for others to be executed safely. A housing-subsidy unwind initiated before the mortgage book has been de-risked transmits a price correction into the banking system through precisely the channel the housing×fiscal interaction describes; the macroprudential framework must therefore precede the subsidy reform. Second, diversification undertaken before the cost base is stabilised imports additional cost-disease into an already-compressing margin structure rather than hedging against it. The sequence below is, in this sense, derived from the risk structure rather than imposed upon it.

5.3 Phase 1 — Stabilising the cost base (the precondition)

The first phase addresses the dominant fiscal trigger and the binding political-economy constraint.

• Wage-indexation reform. As by a wide margin the largest contributor to the structural transition (58.5% Shapley share [M]), the relationship between automatic indexation and sector-specific margin compression is the highest-priority controllable lever. The objective is a mechanism that reduces the volatility transmitted from consumer-price inflation to the sector’s cost base and to indexed public expenditure, recognising that the cost blade is indexed to a price level the revenue blade does not track.
• Pension trajectory. The fiscal–pension clock is a smaller fiscal contributor on the present calibration than the earlier version reported (4.5% Shapley share [M]), in part because the identity that made the pension parameters inert has been repaired and the deficit now begins in 2026 rather than 2028. It must nonetheless be secured, so that reserve drawdown does not become a driver of debt accumulation in its own right. The 2025 reform extends the reserve horizon to approximately 2050 [H]; the actuarial record indicates this is necessary but, against the post-2050 acceleration, not sufficient on its own.
• Housing — macroprudential sequencing. Although housing is a secondary fiscal amplifier (9.2% Shapley share [M]), it is a binding constraint on labour retention and cost competitiveness, and its correction carries financial-stability risk. The macroprudential framework (loan-to-value and debt-service-to-income limits) should be installed before any unwind of demand-side housing support, so that the residual price correction — bounded by the ~13.4% end-2024 overvaluation [H] — proceeds as an orderly adjustment rather than a systemic shock.

5.4 Phase 2 — Capturing the margin (the pivot)

Only once the cost base is stabilised does the pivot from a volume-based to a capability-based model become viable rather than cost-additive. The strategic objective is to ensure that value compressed out of the sector by global fee deflation is captured domestically rather than booked through Luxembourg to external providers. This entails investment in proprietary, AI-native fund-servicing and regulatory-technology infrastructure — the provision of high-complexity, higher-margin services (compliance technology, digital-asset oversight, regulatory tooling) — so that the sector’s value derives increasingly from capability rather than from processing volume. The servicing-value-leakage driver is carried at 0% in the decomposition only because no sourced range could be established for it and it is therefore not varied in the tails (Section 4.4); its qualitative significance for the margin is not thereby measured, and its parameters remain [A].

5.5 Phase 3 — Strategic diversification (long-horizon optionality)

Diversification into adjacent high-value sectors is treated as long-horizon optionality and not as near-term macroeconomic insurance. The model is explicit on this point: the diversification sectors are too small relative to the financial-services base to stabilise the fiscal trajectory within the policy horizon — the job-formation-cost driver contributes 0% to the structural transition on the present calibration [M], again because no sourced range could be established for it, and the cumulative specialist employment generated remains a small fraction of the workforce even in the favourable case. Diversification is therefore most defensibly justified as a source of knowledge spillover to the core economy and as a hedge whose returns accrue beyond the projection horizon, supported by scale-up financing most coherently addressed at the European level. To represent it as a substitute for cost-base stabilisation would be to misstate its quantified contribution.

5.6 Concluding statement

Luxembourg’s trajectory is conditional. The default path — the continuation of current structural trends — is one of managed erosion: a gradual compression of margins and a steady accumulation of debt to a level that, on the central configuration, remains inside the European reference value. This path is not inevitable; the favourable configuration retires public debt entirely and accumulates a net-asset position. Nor is the adverse path inevitable; it is the consequence of the dominant domestic lever being left unaddressed while the remaining drivers stack additively on top of it. The distinguishing variable across all three paths is the same small set of domestically-controllable parameters identified by the variance decomposition. The policy implication is correspondingly precise: stabilise the cost base, then capture the margin, then diversify — in that order, because the order is a property of the risk structure and not a matter of preference. The cost of uncoordinated action is quantified, additive, and domestic; so too is the means of avoiding it.

6. Limitations

The analysis is subject to limitations that bound the interpretation of its results.

Model scope. The model is a strategic decision-support instrument, not an econometric forecast. It incorporates only the behavioural feedbacks explicitly specified — the housing-wealth channel to non-financial value added, the sovereign-financing spread above the debt threshold, and the transmission of indexation to both private compensation and indexed public expenditure. It does not model general-equilibrium labour-market responses, migration dynamics beyond their effect on the parameters, or endogenous monetary-policy reactions. Its outputs are magnitudes and orderings under stated assumptions, not predictions [stated convention].

Residual [A] parameters. Three of the model’s parameters were hardened from [A] to [M] against primary sources (Verification Report): the pension deficit slope (against IGSS/OECD/IMF actuarial projections), the housing-correction depth (against the BCL overvaluation series), and the magnitude of the price-measurement contingency (against the national-accounts identity and ESMA fee data). The servicing-value-leakage parameters remain [A]. Because no sourced range could be established for them they are not varied in the tails, and their Shapley contribution is correspondingly 0% — a consequence of the sourcing discipline rather than a measurement of their unimportance, and a residual uncertainty the decomposition therefore does not bound.

Parameters that remain uncited. The servicing-leakage ceiling is carried from the earlier version without a source and is unchanged [A]. No costing was found for the housing-support measures made permanent after 2025, so no permanent housing increment is modelled; if one exists it would raise the 2040 ratio by roughly three points [A].

The near-term path. The model’s balance from 2027 is below every official estimate. This is disclosed rather than corrected, because no change consistent with the sources closes it and closing it by any other means would be calibration to a target.

One rate for two populations. Financial-sector and whole-economy compensation growth differ, and are now carried as separate parameters. Both are set to 2.9% in the central case on the Ageing Report’s long-run assumption; the realised decade rate for the whole economy is 3.33% and sits in the adverse range.

What the tails do not cover. Drivers without a sourced range are not varied, so the reported spread is a floor on the uncertainty rather than a bound.

The hours assumption. The sector-hours growth rate (Ḣ = +1.5%/yr [M]) is the softest input to the Section 2 decomposition and is load-bearing for the magnitude of the price-measurement contingency. Its conversion from [M] to [H] requires a historical decomposition of AuM growth against sector hours, identified as the principal outstanding item in the verification queue.

The deflator contingency. The weight the paper can place on its challenge to the efficiency interpretation of measured productivity depends on the STATEC deflation methodology for NACE K64.3/K66.3, which is not publicly documented at the required granularity. The analysis is constructed to be robust to either resolution — the structural conclusions rest on the nominal cost side, which is independent of the deflator — but the resolution determines the share of measured decline attributable to price measurement versus genuine efficiency. A formal methodological inquiry to STATEC has been prepared to resolve this contingency. It is important to delimit what the pending resolution affects: it bears solely on the magnitude of the price-measurement contingency in Section 2 — the share of measured productivity decline attributable to price measurement versus genuine efficiency. It does not alter the structural conclusions, which rest on the nominal cost side; it does not affect the policy sequencing of Section 5, which is derived from the risk structure; and it does not affect the Shapley variance decomposition of Section 4.4, which is computed entirely from the fiscal-driver parameters and is independent of the deflator question. The paper is therefore publication-defensible on its present footing, with the STATEC reply serving to harden one parameter from [M] to [H] rather than to validate the analysis.

External baseline divergence. The medium-term trajectory of this paper is more adverse than the authorities’ debt-sustainability analysis, which holds debt near 26% of GDP through 2038, and runs above the IMF’s 2026 Article IV baseline throughout, by a margin of a few percentage points, the two agreeing on direction and mechanism while differing on whether financial-sector tax buoyancy erodes cyclically or structurally. As set out in Section 4.6, the divergence from the authorities’ path is attributable to the assumption that revenues grow with nominal GDP, which this paper quantifies as a cumulative revenue shortfall of approximately 23 percentage points of output by 2040; the divergence from the IMF after 2031 rests on treating the erosion of financial-sector tax buoyancy as structural rather than cyclical. It bears emphasis that the IMF judges the debt itself sustainable and at low risk of sovereign stress [H]; this paper’s concern is the structural margin squeeze and competitiveness, not solvency, and its results should be read as a structural stress-test rather than a competing point forecast.

Channels identified by the IMF but not modelled here. Three channels the 2026 Article IV emphasises are acknowledged rather than formalised, and bound the analysis. (i) Housing as a competitiveness drag. The IMF treats high housing costs as raising reservation wages and eroding competitiveness [H, ¶37]; in the framework here that operates upstream of the cost blade, a cost-side root the Shapley decomposition (which scores housing only as a direct fiscal driver) does not capture. (ii) Exogenous shocks to the revenue blade. The Fund’s risk matrix flags a disorderly AI correction transmitting through fund outflows and lower bank fee income [H, Annex III], and notes that bank profitability is sensitive to investment-fund flows [H, ¶27]; these act on the price term of the Section 2 identity as tail variance around the secular decline modelled here, while an AI-acceleration scenario (≈2% five-year TFP gain, almost double the European average [H, Annex VI]) would act on the throughput term. The asymmetry — exogenous risk falling predominantly on the price-taking revenue blade — reinforces the sequencing rationale of Section 5. (iii) Productivity reading. The IMF attributes the productivity decline to weak R&D and slow digital adoption [H, ¶34], a genuine-efficiency reading; this paper does not contest that such factors operate but argues (Section 2) that a component of the measured decline is a price illusion, a hypothesis the pending STATEC deflator inquiry is designed to discriminate. None of these alters the recurrence engine or the Shapley result; each marks a boundary of the present model and a direction for extension.

7. What changed, and why it matters

This version differs from the June 2026 paper in eleven parameters and two accounting identities. The changes run in both directions and very nearly cancel, which is itself the result worth reporting.

Five commitments and two repaired identities raise the 2040 ratio: the single-tax-class reform, the family and childcare package, a stock-flow adjustment that both official baselines carry, a corrected interest anchor, a pension cash deficit that begins in 2026 rather than 2028, and the repair of an identity under which drawing the pension reserve retired gross debt — an identity that had made every pension parameter in the model inert.

Sourcing the two largest uncited parameters lowers it, and by more. Compensation growth was 3.6% against a realised sector rate of 2.94% and an official long-run assumption of 2.9%. The indexed share of primary expenditure was 0.80 against a supportable range of 0.585 to 0.755. Two further corrections lower it again: the defence ramp was steeper than the government’s own published path, and the house-price correction the earlier version projected forward had already occurred.

The net movement is downward, from the earlier version’s central endpoint of 67.0% to 56.5%. The earlier figure was not wrong by accident and the new one is not right by luck. Both are the arithmetic of their assumptions, and the difference between them is the difference between assumptions that were partly assumed and assumptions that are sourced. A reader who wants to know how much of a projection is evidence and how much is calibration now has the means to check, in changes.json and ranges.json, which record every parameter’s value, its source and its confidence tag.

Appendix — Sources and verification status

Primary sources. European Commission, SWD(2026) 216 (3 June 2026); CSSF UCI statistics; ESMA Costs and Performance of EU Retail Investment Products (2025–26); IGSS / OECD 2024 Ageing Report; IGSS Cahier statistique n°18 (July 2024); OECD Economic Surveys: Luxembourg 2025; IMF Article IV and Technical Assistance Report on Pension Projections (2025); Banque centrale du Luxembourg Financial Stability Review 2025; Conseil national des finances publiques (CNFP), Évaluation des finances publiques (March 2024) and Évaluation du rapport d’avancement annuel du PBSN 2025–2029 (May 2026); International Monetary Fund, Luxembourg: 2026 Article IV Consultation, Country Report No. 26/159 (June 2026); STATEC national accounts; Eurostat nama_10_gdp, nama_10_lp_ulc and gov_10a_main; Eurostat Handbook on Price and Volume Measures in National Accounts (ESA 2010 edition); EDP notification of 7 April 2026; European Commission Spring 2026 forecast.

Confidence migration (per Verification Report). Price-measurement contingency: [A] → [M] (magnitude); applicability pending STATEC inquiry. Pension deficit slope and cliff: [A] → [M], with underlying dates [H]. Housing-correction depth: [A] → [M]. Compensation growth and the indexed share of primary expenditure: [A] → [M]/[H], sourced for the first time in this version (Sections 3.1 and 3.2). Outstanding for full hardening: the AuM/hours decomposition (Ḣ, [M]→[H]); the STATEC deflator confirmation (contingency applicability, [A]→[H]); the servicing-leakage ceiling, which remains [A] and unranged; regional housing disaggregation (enrichment).

Reproducibility. All quantitative results are reproduced from the project’s validated recurrence-relation model and are identical across the working paper, the Verification Report, and the interactive dashboard. The variance decomposition of Section 4.4 is a Shapley attribution over eight driver blocks, computed across all 256 coalitions, verified to sum to the full 110.7 pp Central-to-Adverse transition. The macro and revenue environment block is the residual: every parameter not named in the other seven, namely g_aum, e_fs, g_rest, defl, hp_rec, rate_th, jobs0, vaj, pubsh and taxj. Every parameter’s value, source and confidence tag is recorded in changes.json and ranges.json.

— End of paper —