# Step 247: where the productivity gain goes - prices, wages or profits

**Why.** In the grid the nominal wage per worker is fixed, so "pass-through to prices" lowers the
price level itself: at full pass-through the price index ends at 0.80 at the median and 0.70 when
productivity doubles, about 3% deflation a year for a decade. Under an inflation-targeting central
bank the same gain would appear as higher nominal wages at a stable price level. A referee would ask
whether the housing and fiscal results come from passing the gain on, or from deflation.

**What.** The model already has the switch (`wage_gain`, coupled_thesis_v71): the nominal wage per
employed worker rises with automation progress. Four regimes on the 192 cases of the other grid
factors (v98, no-credit variant, capital feedback 0.3 as in the grid):

| Regime | Wage gain | Pass-through | Price index, year 20 (median) |
|---|---|---:|---:|
| Prices (grid, full pass-through) | 0 | 1 | 0.80 |
| Half to wages | g/2 | 1 | 0.90 |
| Wages (stable prices) | g | 1 | 1.00 |
| Profits (no pass-through) | 0 | 0 | 1.00 |

Generated by `scripts/wage_gain_v247.py`; archived in `data/processed/wage_gain_v247.json`.
The prices regime reproduces the grid's full pass-through cell (72 completed, 19 preserved,
renters 2 of 18).

## Results, benefit indexed to goods prices (the grid's rule)

| Regime | Completed (of 192) | Both displaced kept 90% | Share | Renters | Outright owners | Fixed mortgage | Reset mortgage |
|---|---:|---:|---:|---:|---:|---:|---:|
| Prices | 72 | 19 | 26% | 2/18 | 8/18 | 6/18 | 3/18 |
| Half to wages | 78 | 24 | 31% | 2/18 | 6/18 | 10/22 | 6/20 |
| Wages | 85 | 23 | 27% | 4/22 | 6/22 | 8/20 | 5/21 |
| Profits | 64 | 30 | 47% | 8/16 | 8/16 | 8/16 | 6/16 |

**1. The housing channel is mostly a deflation effect.** With stable prices the gap between
renters and outright owners shrinks from 33 points (11% against 44%) to 9 points (18% against 27%).
Nominal rent and mortgage payments only bite against a benefit that falls with the price level.

**2. The fiscal channel does not depend on deflation.** Whether the gain goes to prices or to wages,
the displaced do worse than when it stays in profits (26%-31% against 47%). Passing the gain on moves
it out of profits, which are taxed at 25%-35% of the share within reach, into either consumer surplus
(untaxed) or wages (taxed at 20%). In all regimes the budget binds at year 10 in the median completed
path (benefits paid equal the room left after existing spending and interest).

**3. The relative position result does not depend on deflation either.** In the matched controls
available here (16 per regime), displaced workers end 24%-28% below the control in every regime, while
retained workers end 25% (prices) to 37% (wages) above it. With stable prices and rising wages the
gap between retained and displaced is wider, because a benefit indexed to goods prices does not share
in wage growth.

## Benefit following nominal output (step 203 rule)

| Regime | Completed | Preserved | Gap to control, displaced without debt (median) |
|---|---:|---:|---:|
| Prices | 101 | 52 | -4.7% |
| Half to wages | 110 | 51 | -5.4% |
| Wages | 120 | 56 | -6.2% |
| Profits | 88 | 55 | +0.8% |

A benefit tied to output roughly closes the relative gap in all regimes, as in step 203. Its higher
completion counts come from the model's closure (higher transfers reduce the residual investment
firms must finance) and are not evidence that the rule is cheaper; see step 248.

## What changes in the text

- "Pass-through hurts renters and mortgage holders" should read "deflation hurts them": the housing
  penalty is a property of a falling price level with nominal housing contracts and a goods-indexed
  benefit, and largely disappears when the gain shows up as higher wages.
- The fiscal result is the robust one: moving the gain out of taxed profits shrinks the room for
  benefits, whether it goes to consumers or to workers.

## Limits

- Rent and mortgage payments stay nominal in all regimes; with wage growth rents would likely rise.
- Completion counts are still set by the stop rules (government cannot borrow, residual investment);
  step 248 replaces them.
- 192 cases per regime, matched controls only where the control path completes.