# Step 305: completing the frontier figure

**Why.** The interactive figure on the thesis page plots the median breakeven productivity gain g\* against
the ceiling on the profit levy. Two sets of points were missing from the archives: the 30% ceiling under
the step-301 costs, and a tax-base erosion elasticity of 0.9, the stronger value tested in step 304. This
step computes them with the existing model and sweep; nothing else in the model changes.

Generated by `scripts/frontier_v305.py`, with output in `data/processed/frontier_v305.json`. The figure data
are rebuilt from it by `scripts/frontier_web.py`. Model: `scripts/closure_model_v304.py`. With the new
options off it reproduces steps 301–303, and the costless and step-301 rows below reproduce steps 277 and
301 exactly. Illustrative; not a forecast.

## Method

- **Sweep.** The step-277 sweep over 144 combinations of the other assumptions: g from 0 to 1.5 in steps
  of 0.025. g\* is the first gain at which both displaced cohorts keep 90% of their consumption.
- **Rules.**
  - PV equity with no levy (0% ceiling);
  - levy up to 15% (the central specification);
  - levy up to 30%;
  - benefits only (step 271 rule).
- **Variants.**
  - costless levy;
  - step-301 costs: erosion 0.5, owners consume 0.4, Tobin's q after the levy;
  - erosion 0.9 with the eroded base returned to owners (as in steps 301–303);
  - erosion 0.9 with the eroded base leaving the economy (step 304's leakage).

## Results

Median g\* (quartiles), and cases preserved out of 432 in the central grid, with fiscal crises in brackets:

| Variant | No levy (PV) | Levy up to 15% | Levy up to 30% | Benefits only |
|---|---|---|---|---|
| Costless levy | 0.463 (0.344–0.625); 214 (110) | 0.325 (0.20–0.45); 277 (10) | 0.20 (0.125–0.30); 338 (56) | 0.50 (0.35–0.675); 211 (110) |
| Step-301 costs | 0.338 (0.175–0.50); 282 (49) | 0.25 (0.125–0.325); 339 (0) | **0.15 (0.10–0.20); 383 (0)** | 0.35 (0.175–0.525); 277 (43) |
| Erosion 0.9, returned to owners | 0.463; 214 (110) | **0.375 (0.275–0.506); 246 (37)** | **0.35 (0.275–0.425); 262 (5)** | 0.50; 211 (110) |
| Erosion 0.9, leaking out | 0.463; 214 (110) | **0.375 (0.275–0.506); 245 (37)** | **0.375 (0.30–0.463); 254 (4)** | 0.50; 211 (110) |

New values in bold. Erosion does not affect the rules without a levy.

**What it shows.**

- **Under the step-301 costs, a higher ceiling still lowers the threshold.** It goes from 0.25 to 0.15.
  The levy's ranking does not change, although all thresholds are lower because owners who consume less
  raise investment in this closure (see step 301).
- **Strong erosion flattens the frontier.**
  - At elasticity 0.9 the 15% levy lowers the median threshold from 0.46 to 0.375, rather than to 0.325.
    Doubling the ceiling to 30% adds little: 0.35, or nothing at all (0.375) when the eroded base leaves
    the economy.
  - The levy still preserves more cases than no levy (246 or 245 against 214), and with far fewer fiscal
    crises (37 against 110).
- **A higher ceiling buys stability more than preservation under strong erosion.** At 30%, crises fall to
  4–5 while preserved cases rise only to 254–262.

## Limits

- Medians over a grid chosen by the researcher; the counts are not probabilities.
- Erosion is a reduced-form response of reported profits to the levy rate. An elasticity of 0.9 lies above
  the estimates the thesis cites (0.13–0.58).
- Same model limits as steps 301–304.