# Stage 113 — Purchasing-power sensitivity frontiers

## Main result

Preserving spending relative to a declining control is not preserving its initial level. In the external-financing control, the minimum annual consumption of the two displaced cohorts is respectively **64.12% and 50.67% of control year 1**. No tested tax case preserves at least 90% of that year-1 anchor in every year for both cohorts.

This does not prove that automation necessarily reduces purchasing power. It is a conditional result for the frozen source trajectory, housing tenure and financing experiment inherited from stages 108–110.

## Declared design

Before execution, the script specifies common q values 0, 0.05, …, 1; three household tax allocations; quarter-end or December collection; and floors 90%, 95%, 97.5%, 99%. There are 126 household-tax cases plus 42 matched external-financing controls.

Both displaced cohorts must meet a floor. Three metrics remain separate:

1. Total twenty-year real consumption divided by total matched-control consumption.
2. Minimum of the twenty annual consumption ratios against the same year's matched control.
3. Minimum annual consumption divided by matched-control year-1 consumption.

The year-1 anchor is **not a pre-automation baseline**. The cumulative metric is an undiscounted sum, not welfare. Floors are declared research thresholds, not standards of living.

Each candidate must also pass the inherited surcharge receipt-schedule gate. This gate checks financing receipts needed by stage 108, not full government or macroeconomic sustainability. Unsupported cases remain in the archive.

## Discrete frontier at 99% of matched control

| Collection | Tax allocation | Passing q: cumulative floor | Passing q: every-year floor |
| --- | --- | --- | --- |
| Quarter-end | Owners only | 0–1 | 0–1 |
| Quarter-end | Per capita | 0–1 | 0–0.90 |
| Quarter-end | Displacement cohorts | 0–0.25 | 0–0.10 |
| December | Owners only | 0–1 | 0–1 |
| December | Per capita | 0–0.20 | 0–0.15 |
| December | Displacement cohorts | None | None |

Ranges summarize **tested points spaced by 0.05 only**. They are not continuous break-even estimates or calibrated parameter bounds. The JSON includes every passing point; no interpolation or monotonicity assumption is used.

For December/displacement-cohort taxation, q ≤ 0.70 fails the receipt schedule (15 cases). Higher tested q passes that gate, but none passes even the 90% matched-control floor for both cohorts. Thus this cell combines funding failure at some assumptions and spending failure at others; it is not a single uniform mechanism.

Across all 126 cases, 111 pass the receipt gate. Taxation of owners only preserves displaced-cohort spending relative to the control under this prototype's assumptions; that does not make it welfare-optimal, empirically validated, or sufficient to preserve initial purchasing power.

## Reproduction and validation

Run:

```sh
python scripts/purchasing_power_frontier_v113.py --output data/processed/purchasing_power_frontier_v113.json
python scripts/test_purchasing_power_frontier_v113.py
```

Six new tests cover a temporary loss hidden by cumulative totals, mandatory funding exclusion, both-cohort acceptance, grid coverage and cash identity, nested floors and archive reproduction. Existing research, web and frontend CI remain required.

[Structured results](data/processed/purchasing_power_frontier_v113.json) retain annual spending, ratios, funding gaps, liquidity shortfall counts, controls, 72 frontier summaries and source hashes. The stage-110 simulator is unchanged; the new script imports it. No estimated coefficients were added.

## Limits and next research step

Common q alone does not represent group heterogeneity. Prices, external income and dividends follow the fixed source path; there is no new macro feedback, labor-market estimate or empirical calibration. Reported ratios are cohort aggregates with fixed masses, not population deciles.

The next priority is to decompose the declining control: income and transfers, prices, debt service, maintenance and consumption. Explain why the control itself misses the year-1 floor before changing policy parameters or extending the frontier to heterogeneous q. That decomposition directly addresses the thesis mechanism rather than treating a relative comparison as proof of preservation.