# Step 304: robustness of the levy conclusion

**Why.** An independent review of steps 301–303 made six points:
- the explanation given for why levy-financed demand support stops helping was wrong;
- the cost channels were tested only up to mid-range values;
- in the model, an eroded tax base returns to owners rather than being lost;
- the test of owners' consumption was mixed with a recalibration of investment;
- the wage curve ignored lay-offs in the demand closure and never passed to prices;
- partial wage indexation with a floor was recommended without being tested.

This step tests each point.

**Code and data.**
- The model is `scripts/closure_model_v304.py`, a copy of step 303. With the new options off it reproduces step 303 exactly (tested).
- The runner is `scripts/robustness_v304.py`; results are in `data/processed/robustness_v304.json`.
- Money is conserved in every run.

**Setup.** All runs use the 432 cases and the revised specification of step 281. The demand closure has symmetric prices, Tobin's q investment and the step-283 interest-rate rule. Step-301 conventions apply unless stated: when owners consume less, autonomous investment is recalibrated, and the slump is kept the same size in points of GDP.

**Terms used below.**
- "Levy effect" is the number of cases preserved with a profit levy of up to 15%, minus the number preserved with present-value equity and no levy.
- Notation: ε is tax-base erosion, ξ the investment tax elasticity, β_w the wage curve, θ the wage-indexation share.

Illustrative; not a forecast.

## 1. Why levy-financed demand support stops helping

Support here means levy-financed public spending at any demand gap: φ = 1, up to 5% of output, under a shared 15% levy ceiling. Its effect is the number of cases preserved with support minus without. This effect was computed for all 16 combinations of four cost channels:
- E: erosion ε = 0.5
- O: owners consume 0.4
- Q: Tobin's q computed after the levy
- X: investment tax elasticity ξ = 1.0

Shapley values split the change between no costs and all four channels:

| Decay | Support effect, no costs | With all four | E | O | Q | X |
|---:|---:|---:|---:|---:|---:|---:|
| 0 | −8 | −48 | −2.0 | −18.5 | −10.0 | −9.5 |
| 1.0 | −12 | −91 | −8.8 | −23.4 | −12.6 | −34.2 |
| 1.27 (stress) | +16 | −74 | −1.4 | −45.8 | −7.4 | −35.4 |

**What it shows.**
- **Support helps only in the stress case.** With no costs it already hurts at decays 0 and 1.0. It helps only at 1.27.
- **The reversal comes mainly from owners' consumption and the investment response, not from erosion.**
  - Owners' lower propensity to consume, in the recalibrated economy, contributes about half of the swing at 1.27.
  - Investment responding to after-tax returns contributes about 40%.
  - The two reinforce each other. At 1.27 support still helps with O alone (+15) or X alone (+11), but hurts with both (−37).
  - Erosion contributes little.
- **Correction to steps 301–303.** Their text said support stops helping "because the levy that finances it lowers investment". That is part of the story. The investment response matters mainly together with owners who save more.

## 2. Stronger costs, thresholds and fiscal crises

Levy effect and fiscal crises (peak net debt above 95% of output), central specification:

| Erosion ε | Preserved: levy / no levy | Levy effect | Fiscal crises: levy / no levy |
|---:|---|---:|---|
| 0 | 277 / 214 | +63 | 10 / 110 |
| 0.5 | 260 / 214 | +46 | 16 / 110 |
| 0.9 | 246 / 214 | +32 | 37 / 110 |
| 2.0 | 220 / 214 | +6 | 77 / 110 |
| 5.0 | 212 / 214 | −2 | 181 / 110 |

Demand stress case (decay 1.27):

| Spec | Preserved: levy / no levy | Levy effect | Fiscal crises: levy / no levy |
|---|---|---:|---|
| No costs | 131 / 60 | +71 | 290 / 290 |
| ε = 0.9 | 81 / 60 | +21 | 290 / 290 |
| ξ = 1.6 | 101 / 53 | +48 | 298 / 300 |
| ε = 0.9, ξ = 1.6 | 71 / 53 | +18 | 298 / 300 |
| Step-301 costs, ξ = 1.6 | 227 / 198 | +29 | 147 / 127 |
| Same, no slump (decay 0) | 288 / 286 | +2 | 109 / 99 |

**Thresholds.** These are the values at which the levy effect reaches zero, found by bisection with the other channels off:
- **Erosion ε\*:** about 2.4 in the central specification and about 1.9 in the stress case. The literature range used in step 301 is 0.13–0.58, with larger estimates in some studies.
- **Investment elasticity ξ\*:** not reached up to 10, with or without the step-301 costs.

**What it shows.**
- **The levy effect survives stronger costs** (ε 0.9, ξ 1.6). It shrinks to near zero only at an erosion elasticity about four times the upper end of the estimates used.
- **It is a trade-off, not a free win.**
  - In the central specification, the levy has fewer fiscal crises than no levy at least up to ε = 2 (77 against 110), and more at ε = 5 (181 against 110); values in between were not tested.
  - With the step-301 costs and ξ = 1.6, it has more crises than no levy: 147 against 127 in the stress case, and 109 against 99 without a slump.
  - Without a slump, its advantage in preserved cases is then only +2.

## 3. Erosion as leakage

In steps 301–303 the eroded part of the profit base stayed with firms and reached owners. Here, as a variant, it leaves the economy (profit shifting abroad). It is recorded in an external account, so the money identity still holds.

| Spec | Transfer: levy effect (crises) | Leakage: levy effect (crises) |
|---|---|---|
| Central, ε = 0.5 | +46 (16) | +46 (16) |
| Central, ε = 0.9 | +32 (37) | +31 (37) |
| Stress case, ε = 0.5 | +33 (290) | +33 (290) |
| Stress case, ε = 0.9 | +21 (290) | +17 (290) |

The erosion threshold ε\* falls from about 2.4 to about 2.0 with leakage. Treating the eroded base as a transfer to owners was not what kept the levy effective. The government grosses up the levy to collect what it needs, and the ceiling binds only in some cases, so the destination of the eroded amount matters little.

## 4. Owners' propensity to consume, holding the aggregate

At year 0, owners account for 25%–27% of household spending under broad ownership and 51%–53% under concentrated ownership. To lower owners to 0.4 while keeping the aggregate propensity at 0.9, the others would have to consume 107%–144% of their income. That is not feasible.

The feasible aggregate-preserving split is others at 1.0 and owners at 0.6 (broad) or 0.8 (concentrated). The table reports levy effects (gaps), not levels:

| Spec | Central | Decay 0 | Decay 1.0 | Decay 1.27 |
|---|---:|---:|---:|---:|
| Owners 0.9 (baseline) | +63 | +61 | +84 | +71 |
| Owners 0.4, step-301 recalibration | +68 | +73 | +76 | +67 |
| Owners 0.4, no recalibration | — | +57 | +23 | +15 |
| Aggregate-preserving split | +74 | +62 | +98 | +56 |

**What it shows.**
- **The levy effect is positive under every calibration.**
- **The levels depend heavily on the calibration.**
  - Without recalibration, owners at 0.4 leave the economy in a deep shortfall: 28 of 432 preserved with the levy at 1.27, against 293 with recalibration.
  - Comparisons of levels across these variants therefore say little about the levy. The gaps are the relevant measure.
- **In the central specification**, the rise in levels when owners save more comes from investment being whatever is not consumed: saving becomes capital by construction.

## 5. Wage curve with Okun lay-offs, and wage cuts passed to prices

Two variants are added to the β_w = 0.1 wage curve:
- **Okun:** the wage curve also responds to Okun lay-offs (okun_wage_curve = 0.1).
- **ψ:** a share ψ of the wage saving passes to lower prices.

Gaps are median differences from the status-quo control, mean of the last five years.

| Spec | Preserved: levy / no levy | Levy effect | Gap: retained workers; displaced (no debt / debt) |
|---|---|---:|---|
| Central, fixed wage | 277 / 214 | +63 | +28.9; +2.8 / +4.0 |
| Central, β_w 0.1 | 298 / 232 | +66 | +14.1; +2.7 / +3.9 |
| Central, β_w 0.1, ψ 0.5 | 288 / 229 | +59 | +13.5; +1.6 / +2.8 |
| Central, β_w 0.1, ψ 1 | 275 / 213 | +62 | +15.1; +1.0 / +2.0 |
| Decay 1.27, fixed wage | 131 / 60 | +71 | +4.9; −24.8 / −28.1 |
| Decay 1.27, β_w 0.1 | 144 / 65 | +79 | −6.8; −21.4 / −25.4 |
| Decay 1.27, β_w 0.1 with Okun | 144 / 66 | +78 | −7.8; −20.9 / −24.7 |
| Decay 1.27, with Okun and ψ 1 | 119 / 57 | +62 | −4.5; −27.2 / −33.1 |

**What it shows.**
- **Okun lay-offs add little to the wage curve.** The extra unemployment they create is small next to the imposed 20%.
- **Without pass-through, the wage curve looks like it "strengthens" the levy. That is a redistribution.**
  - The retained workers' gap to the control falls from +29% to +14%, and owners' rises from +43% to +55%.
  - The wage saving goes to profits and to the levy base.
- **When the saving passes fully to prices**, the levy effect returns to about its fixed-wage size (+62), and the displaced gain nothing from the wage curve.

## 6. Wage indexation with a floor, and an 80% replacement rate

Two preservation measures are reported:
- **Original:** 90% of pre-automation consumption.
- **Indexed:** 90% of the indexed reference.

Cost is the median benefit cost as a share of output.

| Wages | Benefit | Preserved: original / indexed | Levy effect: original / indexed | Cost | Gap: displaced (no debt / debt) |
|---|---|---|---|---:|---|
| Fixed (ω 0) | fixed | 277 / 277 | +63 / +63 | 3.8% | +2.8 / +4.0 |
| Fixed | θ 0.5 + floor | 303 / 292 | +67 / +63 | 4.2% | +6.4 / +8.1 |
| ω 0.5 | fixed | 279 / 279 | +67 / +67 | 3.9% | −8.9 / −11.2 |
| ω 0.5 | θ 0.5 + floor | 292 / 273 | +85 / +71 | 5.6% | +10.5 / +7.5 |
| β_w 0.1 | fixed | 298 / 298 | +66 / +66 | 3.8% | +2.7 / +3.9 |
| β_w 0.1 | θ 0.5, no floor | 275 / 307 | +54 / +65 | 3.8% | −2.0 / −0.5 |
| β_w 0.1 | θ 0.5 + floor | 318 / 306 | +66 / +66 | 3.9% | +3.4 / +4.5 |
| ω 0 | 80% replacement, fixed | 122 / 122 | +13 / +13 | 3.0% | −20.7 / −20.1 |
| ω 0 | 80%, θ 0.5 + floor | 176 / 111 | +29 / +15 | 3.3% | −15.2 / −14.6 |
| ω 0.5 | 80%, θ 0.5 + floor | 277 / 152 | +59 / +25 | 4.5% | −2.3 / −3.8 |
| ω 0.5 | 80%, θ 1 + floor | 291 / 145 | +68 / +30 | 5.8% | +12.8 / +12.0 |
| β_w 0.1 | 80%, θ 0.5 + floor | 126 / 125 | +11 / +11 | 3.1% | −17.0 / −16.6 |

**What it shows.**
- **Half indexation with an upward-only floor is safe in both directions.**
  - When wages rise (ω 0.5), the floor never binds. The result equals plain half indexation: it closes the gap at 5.6% of output.
  - When wages fall (β_w 0.1), plain half indexation lowers the benefit, and preservation falls from 298 to 275 on the original measure. The floor prevents this: 318 on the original measure and 306 on the indexed one.
- **The levy effect is smaller on the indexed measure**, about +63 to +71 instead of +67 to +85. The step-303 statement that indexation "makes the levy matter more" holds mainly on the original measure.
- **With an 80% replacement rate** few cases keep 90% of pre-automation consumption: 122–125 with a fixed benefit. Indexation helps when wages rise (277 at ω 0.5 with half indexation and a floor). It does not help when wages fall.

## Limits

- **The decomposition uses the step-301 conventions.** The owners' channel includes the recalibration of autonomous investment. Section 4 shows that the levels, but not the sign of the levy effect, depend on this.
- **Thresholds are found by bisection,** which assumes the levy effect falls steadily as the elasticity rises. They are not structural estimates.
- **Leakage is modelled as an outflow to an external account.** It is not a real resource cost, and it has no exchange-rate or trade effects.
- **The pass-through of wage cuts to prices** is a simple proportional rule.
- **The fixed-benefit rule and the 90% threshold are unchanged.** Preservation is still a count over a grid that is not probability-weighted.