# Step 303: a benefit indexed to wages

**Why.** Step 302 found a cost of wage sharing that the levy cannot fix. When a share ω of the automation gain goes
to the wages of the employed, the levy still protects the displaced against a fall in their own living standard.
But the benefit reference is fixed in goods. At ω = 0.5 the displaced end 9%–11% below the status-quo control,
while the employed end about 49% above it. This step asks whether indexing the benefit to wages closes that gap,
what it costs the budget, and whether it changes the levy conclusion.

The model is `scripts/closure_model_v303.py`, a copy of step 301. With its new option off it reproduces steps 301
and 302 exactly (tested). The runner is `scripts/wage_index_v303.py`, with output in
`data/processed/wage_index_v303.json`. Money is conserved in every run. Illustrative; not a forecast.

## The rule

The benefit tops displaced workers' income up to a reference. Until now the reference was their pre-automation
disposable income, fixed in goods. Here it follows the real net wage per employed worker:

    reference_t = reference_0 × (1 + θ × (W_t / W_0 − 1))

- **W** is the wage net of wage tax, deflated by the goods price.
- **θ** is the degree of indexation: 0 is the old rule, 0.5 is half indexation, 1 is full indexation.
- **With a wage curve** (β_w, step 302), wages fall with unemployment, so full indexation also *lowers* the
  benefit. A variant with an upward-only floor (the factor never goes below 1) is reported for that case.

**Two measures of preservation.**
- **Original reference** (the thesis measure): consumption of both displaced cohorts stays at or above 90% of its
  pre-automation level in every year.
- **Indexed reference:** consumption stays at or above 90% of the pre-automation level times the indexation
  factor. It asks whether the indexed promise is kept. For θ = 0 the two measures are the same.

**Fiscal cost.**
- Total benefit cost and total levy as shares of total output over the 20 years (medians across cases).
- Cases using the levy and cases in which the benefit is rationed.

## Results: central specification, 432 cases

The "gap" columns are the median gap to the status-quo control (mean of the last five years). "Levy's effect" is
preservation with the levy (up to 15%) minus preservation with present-value equity and no levy.

| Wages | θ | Preserved: original / indexed | Gap: employed; displaced | Benefit cost, % of output | Levy used / rationed | Levy's effect | Median g\* |
|---|---:|---|---|---:|---|---:|---:|
| Fixed wage (ω 0) | 0 | 277 / 277 | +28.9; +2.8 / +4.0 | 3.8 | 182 / 129 | +63 | 0.325 |
| Fixed wage (ω 0) | 1 | 310 / 289 | +25.2; +9.8 / +10.5 | 4.4 | 202 / 141 | +75 | 0.325 |
| ω 0.25 | 0 | 278 / 278 | +44.9; −0.2 / −0.2 | 3.9 | 195 / 142 | +68 | |
| ω 0.25 | 1 | 301 / 281 | +38.8; +25.0 / +25.8 | 5.8 | 238 / 181 | +99 | |
| **ω 0.5** | **0** | **279 / 279** | **+48.8; −8.9 / −11.2** | **3.9** | **204 / 162** | **+67** | **0.40** |
| ω 0.5 | 0.5 | 292 / 273 | +44.3; +10.5 / +7.5 | 5.6 | 230 / 200 | +85 | 0.375 |
| ω 0.5 | 1 | 289 / 247 | +38.5; +21.0 / +17.0 | 7.1 | 284 / 229 | +116 | 0.388 |
| β_w 0.1 | 0 | 298 / 298 | +14.1; +2.7 / +3.9 | 3.8 | 166 / 102 | +66 | |
| β_w 0.1 | 1 | 245 / 326 | +13.9; −5.2 / −4.6 | 3.7 | 156 / 90 | +44 | 0.30 |
| β_w 0.1 | 1, floor | 322 / 318 | +13.5; +4.3 / +5.6 | 3.9 | 169 / 103 | +69 | |

Under the step-301 costs (erosion 0.5, owners consume 0.4, q after the levy), at ω 0.5:

| θ | Preserved: original / indexed | Gap: employed; displaced | Benefit cost, % of output | Rationed | Levy's effect | Median g\* |
|---:|---|---|---:|---:|---:|---:|
| 0 | 329 / 329 | +43.0; −15.5 / −18.4 | 4.5 | 99 | +35 | 0.30 |
| 0.5 | 344 / 325 | +34.5; +9.9 / +5.3 | 6.1 | 149 | +89 | |
| 1 | 341 / 277 | +33.2; +23.2 / +13.5 | 7.6 | 217 | +100 | 0.325 |

There are no fiscal crises under the step-301 costs. Without them there are 10–14 crises at every θ. In the cases
that use the levy, the median peak levy rate is at the 15% ceiling whatever θ (slightly below it only under the
wage curve with the step-301 costs).

**Demand closure with the step-283 rate rule** (preserved, original / indexed; fiscal crises in brackets):

| Decay | Wages | θ | Levy | No levy |
|---:|---|---:|---|---|
| 0 | ω 0 | 0 | 298 / 298 (25) | 237 / 237 (150) |
| 0 | ω 0 | 1 | 304 / 290 (25) | 233 / 223 (163) |
| 0 | ω 0.5 | 0 | 273 / 273 (18) | 218 / 218 (141) |
| 0 | ω 0.5 | 1 | 259 / 218 (19) | 139 / 130 (198) |
| 1.27 | ω 0 | 0 | 131 / 131 (290) | 60 / 60 (290) |
| 1.27 | ω 0 | 1 | 190 / 149 (300) | 80 / 62 (300) |
| 1.27 | ω 0.5 | 0 | 129 / 129 (276) | 57 / 57 (288) |
| 1.27 | ω 0.5 | 1 | 252 / 188 (342) | 93 / 65 (352) |

**What it shows.**

- **Indexation closes the gap, and half indexation is enough.**
  - At ω 0.5, half indexation turns the displaced's −9% / −11% into +10% / +8% against the control.
  - Full indexation overshoots to +21% / +17%. The employed still end higher (+38% to +44%).
  - The same holds under the step-301 costs.
- **It roughly doubles the cost of the benefit when wages rise.** At ω 0.5 the benefit costs 3.9% of output
  with no indexation, 5.6% with half and 7.1% with full. The levy is used in more cases (204 → 230 → 284), and
  the benefit is rationed in more cases (162 → 200 → 229), because the levy hits its 15% ceiling.
- **The indexed promise is harder to keep than the old one.** Preservation against the original reference
  rises slightly (279 → 292 → 289). Against the indexed reference it falls (279 → 273 → 247): the benefit is
  promised more than the budget can always pay.
- **The levy matters more, not less, against the original reference.** Its effect grows from +67 to +85 and +116
  cases. Against the indexed reference the increase is much smaller (step 304: +63 to +71 with half indexation). Without the levy,
  indexation in the demand closure leaves 139 cases instead of 218 at decay 0. A promise that grows with wages
  needs a revenue source that grows with the gain.
- **With a falling wage, full indexation backfires.** The benefit falls with wages: 245 cases preserved instead
  of 298, and the displaced end about 5% below the control. An upward-only floor avoids that (322 cases).
- **In a demand slump, indexation helps demand but raises crises.** At decay 1.27 and ω 0.5, preservation
  against the original reference rises from 129 to 252. Fiscal crises rise from 276 to 342.

## Limits

- **The wage index is the model's average real net wage of the employed**, with no lag and no distinction
  between occupations. A real index would follow a published wage statistic with a lag.
- **The replacement rate stays at 100%.** Indexation on top of full replacement is generous by construction.
  Half indexation with a lower replacement rate was not tested here. Step 304 tests half indexation with an
  upward-only floor and an 80% replacement rate, and reports both preservation measures.
- **No behavioural response of the displaced to a more generous benefit:** employment is imposed, as everywhere
  in the thesis.
- **The levy ceiling stays at 15%,** so the extra cost shows up as rationing. A higher ceiling would trade
  rationing for a larger levy (step 276: up to 30%).