# Stage 114 — Why the control loses purchasing power

## Finding

The source trajectory loses wage income faster than transfers, dividends and lower prices replace it. The minimum annual real consumption occurs in **year 10**: **64.12%** of control year 1 for the displaced cohort without personal debt and **50.67%** for the cohort with personal debt. By year 20 these ratios recover to **66.90%** and **53.32%**.

Neither cohort has a cash-constrained consumption month in this external-financing control. Under its inherited 90%-of-current-spendable-income rule, consumption falls as the income available for spending falls. This is conditional accounting evidence about this scenario, not a causal empirical finding.

## Exact decomposition at the trough

Contributions are **percentage points of each cohort's own year-1 real consumption**. They sum to the consumption change, up to rounding.

| Component | Displaced, no personal debt | Displaced, personal debt |
| --- | ---: | ---: |
| Wage change | -104.36 | -127.17 |
| Dividend change | +18.06 | +9.60 |
| Transfer change | +38.65 | +54.86 |
| Contractual personal-interest change | 0.00 | +2.49 |
| Nominal maintenance change | 0.00 | 0.00 |
| Positive-income floor and cash constraint | 0.00 | 0.00 |
| Price effect, including its interaction with nominal spending | +11.78 | +10.89 |
| **Net change** | **-35.88** | **-49.33** |

The wage contribution can exceed 100 percentage points because the denominator is consumption net of costs, rather than gross wages. Comparing the two columns does not isolate a causal debt effect: their reference consumption, dividends and transfer amounts differ.

A zero nominal-maintenance contribution does **not** mean that maintenance has no real burden. Its nominal amount is unchanged; the interaction with price belongs in the price term. Likewise, the positive interest contribution means interest costs fall relative to year 1, not that debt is harmless.

## Method

The script reconstructs nominal wages, dividends and benefits from the exact annual source, using beginning-of-period equity units, the source payout and benefit allocation. Each year's reconstructed sum is checked against the frozen outside income supplied to the monthly control.

Monthly control contractual interest and maintenance enter desired spending; personal principal remains an asset exchange. Positive-income clipping and the difference between desired and realized consumption are retained separately. Principal may affect cash availability but is not subtracted from income a second time.

For each nominal consumption component N_k and inverse price z=1/P, the attribution between year 1 and year t is:

- Nominal component contribution: (N_kt - N_k1) × (z_t + z_1)/2.
- Price contribution: (ΣN_kt + ΣN_k1)/2 × (z_t - z_1).

This symmetric allocation splits the nominal-price interaction equally and adds exactly to the real-consumption change. It is an accounting convention, not a causal intervention. Another interaction convention can change component allocations while preserving the total.

Control year 1 is the anchor; it is not a pre-automation observation. Fixed cohort masses mean the within-cohort retention ratios also apply per cohort member. They are not population-decile estimates.

## Reproduction and validation

```sh
python scripts/consumption_decomposition_v114.py --output data/processed/consumption_decomposition_v114.json
python scripts/test_consumption_decomposition_v114.py
```

Six tests cover pure-price and pure-income changes, interactions and reversal, exact consumption identities, recovery of the stage-113 control, principal classification and archive reproduction. Comparisons of equivalent annual/monthly sums use floating-point tolerance. Existing repository research, web and frontend checks remain the merge gate.

[Annual components and attribution](data/processed/consumption_decomposition_v114.json) include all twenty years, exact source hashes and the maximum source-income reconstruction error. No existing simulation parameters or source equations were changed.

## Implication for the thesis

Lower prices help in this scenario but do not fully offset the loss of labor income. Transfer adequacy therefore needs a level-based test, alongside fiscal feasibility, rather than only a comparison with a declining control.

Next calculate the additional **real-consumption gap** to a declared floor and the corresponding nominal resources. Keep the required resources separate from an implementable transfer: any supplemental payment must have explicit funding, tax incidence and a rerun of household cash flows before it can be called feasible. Do not claim that this accounting decomposition validates the automation hypothesis or identifies the optimal policy.