# F5-conf-04 — Value-function iteration against a closed-form policy (optimal growth with log utility)

## Scientific question
A planner maximises Σ_t β^t ln c_t subject to y_{t+1} = (y_t − c_t)^α, 0 < c_t < y_t, with full depreciation and productivity normalised to one. The Bellman equation is V(y) = max_{0<c<y} { ln c + β V((y − c)^α) }. This model has a known closed-form solution for the optimal consumption policy c(y) and the value function V(y). The estimands are: the closed-form policy at three evaluation points of y; the policy obtained by value-function iteration (VFI) on the prescribed grid at the same points; the maximum absolute discrepancy between them over the evaluation points; the closed-form and numerical value function at y = 1; the savings rate implied by the closed form; and the contraction modulus of the Bellman operator in the sup norm.

## Inputs
- `data/model.csv` — columns `parameter`, `value`, `description`: `alpha`, `beta`, the grid specification (`y_min`, `y_max`, `grid_points`, `grid_spacing` = log, i.e. points equally spaced in ln y), the VFI stopping tolerance `vfi_tolerance` (sup-norm change between successive value functions), and the three evaluation points `eval_y_1`, `eval_y_2`, `eval_y_3`. Theoretical parameters, not observed data.

## Information-availability rules
- Use only `data/model.csv` and solve the model on the prescribed grid with the prescribed parameters; if any deviation from the prescribed design is necessary, record it in `data_issues`.
- The numerical policy must come from the VFI, not from the closed form; the closed form must be derived (e.g. guess-and-verify on the Bellman equation) and stated in the report.

## Required outputs
- `out/results.json` with the keys:

| key | type | unit | meaning |
|---|---|---|---|
| `task_id` | string | — | "F5-conf-04" |
| `c_closed_y05`, `c_closed_y1`, `c_closed_y2` | number | units of output | closed-form optimal consumption at y = 0.5, 1.0, 2.0 |
| `c_num_y05`, `c_num_y1`, `c_num_y2` | number | units of output | VFI consumption policy at y = 0.5, 1.0, 2.0 (interpolated on the grid if the point is not a grid node) |
| `max_abs_error_policy` | number | units of output | max over the three evaluation points of \|c_num − c_closed\| |
| `value_closed_y1` | number | utils | closed-form V(1) |
| `value_num_y1` | number | utils | VFI value function at y = 1 |
| `savings_rate_closed_form` | number | — | closed-form (y − c)/y (the baseline quantity) |
| `contraction_modulus` | number | — | modulus of the Bellman operator as a contraction in the sup norm |
| `grid_points_used` | integer | — | number of grid points actually used |
| `n_iterations` | integer | — | VFI iterations until the stopping rule was met |
| `final_sup_change` | number | utils | sup-norm change at the last iteration |
| `feasibility` | string | — | "complete", "partial" or "infeasible" |
| `feasibility_reason` | string or null | — | explanation if not complete |
| `data_issues` | array of {"issue","handling"} | — | data problems or design deviations and how handled (empty list if none) |
| `n_input_rows` | object | — | {"model.csv": rows read} |
| `n_analysed` | integer or null | — | number of grid points |
| `random_seed` | integer or null | — | seed used, if any |

- `out/report.md` with the sections: Question; Data and exclusions; Method; Results; Uncertainty; Limitations and open issues. State explicitly the assumptions of the closed form (log utility, Cobb–Douglas technology, full depreciation, interior solution), why the Bellman operator is a contraction and with what modulus, and the scope of the numerical validation (which grid, which parameter point, what the discretisation error depends on).
- `out/analysis.py` — running `~/wp29/venv/bin/python out/analysis.py` from the workspace root must regenerate `out/results.json` from `data/` only, deterministically.
- `out/claims.json` — a list of the central claims of the report: [{"id": "C1", "text": "...", "results_keys": ["..."], "status": "supported"|"unsupported"|"not_evaluated", "caveat": "..."}]. Each claim's status must be justified by the listed results keys.

## Constraints
- No network access. Python: `~/wp29/venv/bin/python` (numpy, pandas, scipy, statsmodels available). Do not modify `data/`.
- This task is designed to be completed in well under 20 minutes of work; the VFI on the prescribed grid runs in seconds when vectorised.