# Step 250: what the gap to the control is made of

**Why.** The "relative position" result (displaced workers end below a matched control without
automation) was challenged on three counts: the control keeps each case's tax policy, so only cases
with the status-quo policy had a complete control; the gap mixes several mechanisms; and part of the
grid is duplicated. Generated by `scripts/gap_decomposition_v250.py`; archived in
`data/processed/gap_decomposition_v250.json`. Closures of step 248 (debt cap 90%, firm overdraft)
unless stated.

## 1. Duplicate cells

The stress setting raises interest rates and housing costs for mortgage holders but does not change
rent or outright owners' costs. All 144 renter and outright-owner pairs are identical across stress
settings (checked). The grid has **432 distinct cases**, not 576; counts below use the 432.

## 2. A control without policy change

| Closure | Control | Matched | Both displaced below | Retained workers above | Median gap, displaced (no debt / debt) |
|---|---|---:|---:|---:|---|
| Strict | same policy (grid) | 21 | 100% | 100% | -25% / -21% |
| Strict | status-quo policy | 51 | 100% | 100% | -22% / -17% |
| Closures | same policy | 432 | 91% | 96% | -24% / -20% |
| Closures | status-quo policy | 432 | 93% | 94% | -23% / -20% |

The choice of control hardly matters. Under strict rules every matched pair has the displaced below
the control; with closures 91%-93% do.

## 3. Decomposition (all 432 distinct cases)

Switches applied cumulatively; numbers are the change in the final consumption index of the
displaced (average of the two cohorts, initial level = 100), mean over cases (median in brackets):

| Component | Mean | Median |
|---|---:|---:|
| Gap to the status-quo control, baseline | -39.2 | -25.4 |
| of which: growth of the same cohorts in the control (employed) | 13.7 | 10.3 |
| R: no rationing of the benefit (no debt cap) | +22.9 | 0.0 |
| D: no claw-back of dividends on the cohort's own savings | +31.9 | +28.8 |
| H: benefit indexed to goods and housing | +4.9 | +2.5 |
| **Gap after all three switches** | **+20.5** | **+15.1** |

After the three switches the displaced end above the control in 75% of cases.

**What it means.**
- **Rationing** matters in a minority of cases (median 0) but is large where the budget binds. Removing
  it needs net debt above 90% of output in 37% of cases, up to 2.7 times output at the 90th percentile.
- **The claw-back is the largest single component.** The benefit tops income up to the reference and
  subtracts the cohort's own dividends one-for-one. The displaced keep saving out of the benefit, buy
  equity, and in a growing, automating economy those dividends rise; the rule withdraws all of it, a
  100% marginal tax on their capital income. Limiting the claw-back to the dividend already in the
  reference income lifts the displaced well above where they would otherwise end.
- **Housing indexation** adds a little on top, more under full pass-through (step 247: the housing
  effect is a deflation effect).
- The control's growth is real but smaller than the gap.

**So the relative-position result is a property of the benefit design, not a law of the model.** A
benefit set against today's standard, rationed when the budget binds and withdrawn against the
recipient's own capital income, leaves the displaced behind. A benefit that lets them keep the returns
on their own savings does not, if the budget can carry it.

## Limits

- The switches are applied in one order; contributions in another order would differ.
- Removing rationing ignores the debt it requires; there is no interest premium in the model.
- The model's saving rule (90% of income consumed, the rest invested) drives the claw-back effect.