# Step 277: breakeven productivity instead of shares of cells

**Why.** The third referee report showed that the grid's preservation count samples a threshold
coarsely. Along the productivity gain, the outcome is close to a step: either the debt cap binds and the
benefit is rationed, or it does not. So "49% of cells" mostly counts cells on one side of a threshold. The
informative result is the threshold itself.

Generated by `scripts/breakeven_v277.py`; archived in `data/processed/breakeven_v277.json`. Model of
step 276. Illustrative; not a forecast.

## Method

- **Sweep.** For each of the 144 distinct combinations of the other factors, the productivity gain g is
  swept from 0 to 1.5 in steps of 0.025.
- **Breakeven gain.** g\* is the smallest gain at which both displaced cohorts keep 90% of their initial
  consumption in every year.
- **Government rules.** Three are compared:
  - benefits only: the step-271 rule, in which only the benefit is cut when the 90% cap binds;
  - revised: present-value equity, plus a profit levy of up to 15% of profits when the cap binds
    (step 276);
  - revised with a levy of up to 30%.

**What g means in output.** g is not total-factor productivity. It is the efficiency gain in a scenario
that also displaces 20% of workers. Real output at year 20, relative to the no-automation control
(pass-through 0.5, broad ownership, renters, revised rule):

| g | 0 | 0.1 | 0.2 | 0.25 | 0.3 | 0.4 | 0.5 | 0.75 | 1 |
|---|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| Output vs control | 0.63 | 0.74 | 0.81 | 0.85 | 0.89 | 1.04 | 1.22 | 1.49 | 1.74 |

At g ≈ 0.38 the automation scenario produces as much as the economy without automation. Below that,
displacing 20% of workers lowers output.

## Results

| Rule | Median g\* (quartiles) | Output at the median g\* | Never preserved (of 144) | Preserved at g = 0.25 | at g = 0.4 | at g = 0.5 |
|---|---|---:|---:|---:|---:|---:|
| Benefits only | 0.50 (0.35–0.68) | 1.22 | 7 | 14% | 33% | 44% |
| **Revised (levy 15%)** | **0.33 (0.20–0.45)** | **0.913** | 6 | 29% | 61% | 73% |
| Revised, levy 30% | 0.20 (0.13–0.30) | 0.81 | 5 | 56% | 76% | 81% |

"Preserved at g" counts combinations with g\* at or below g that stay preserved at every higher gain.

Median g\* by factor, revised rule:

| Factor | Values | Median g\* |
|---|---|---|
| Tax reach | 0.5 / 1 | 0.43 / 0.20 |
| Tax rate | 25% / 35% | 0.35 / 0.30 |
| Ownership | concentrated / broad | 0.39 / 0.30 |
| Pass-through | 0 / 0.5 / 1 | 0.30 / 0.33 / 0.36 |
| Tenure | renters / outright owners / fixed / reset mortgage | 0.35 / 0.34 / 0.30 / 0.35 |

**Not monotone.** In 15 combinations under the revised rule, preservation is lost again at higher gains.
These are full pass-through with concentrated ownership. A larger gain lowers prices further, and nominal
rent and mortgage payments do not fall with them (step 272).

**How much tax is needed.** The smallest levy cap that preserves the displaced, revised rule, by
productivity gain:

| g | Median levy cap needed (quartiles) | Not possible with a levy up to 50% |
|---|---|---:|
| 0.10 | 30% (20%–45%) | 86 of 144 |
| 0.25 | 17.5% (5%–29%) | 28 |
| 0.40 | 7.5% (0%–17.5%) | 12 |

## What it shows

- **With benefits as the only margin, protection requires automation that raises output well above the
  economy without it.** The median breakeven gain implies output about 21% higher.
- **If the profit tax may rise by up to 15% of profits when the budget binds,** the breakeven falls to a
  gain at which output is about 9% lower than without automation (0.913 in the reference case, computed with `breakeven_v277.output_ratio(0.325)` and archived in `data/processed/breakeven_median_output_v277.json`). With 30%, about 19% lower.
- **The answer to the research question is therefore conditional on two numbers:** how much automation
  raises output, and how far the profit tax may rise. The breakeven surface gives the trade-off directly.
  In the median combination, an automation that cuts output by about a quarter (g = 0.1) needs a levy
  cap of 30%. One that cuts it by about 15% (g = 0.25) needs 17.5%. One that leaves output roughly
  unchanged (g = 0.4) needs 7.5%.
- **Tax reach is the strongest institutional lever.** Full reach lowers the breakeven gain from 0.43 to
  0.20.